INTRODUCTION
TO M O D E R N
MATHEMATICS
H E L E N A RASIOWA
University of Warsaw
1973
NORTH-HOLLAND
PUBLISHING
A M S T E R D A M · L O N D O N
COMPANY
AMERICAN ELSEVIER PUBLISHING COMPANY,
NEW
YORK
INC.
Translated by
Olgierd Wojtasiewicz
Copyright 1973 by
PANSTWOWE WYDAWNICTWO NAUKOWE
(PWN — POLISH SCIENTIFIC PUBLISHERS)
Warszawa
All rights reserved
This book is a translation from the original Polish
Wstep do matematyki wspolczesnej
published by PWN — Polish Scientific Publishers, 1971
in the series "Biblioteka Matematyczna"
The English edition of this book has been published by PWN jointly with
NORTH-HOLLAND PUBLISHING COMPANY
Amsterdam
ISBN 0 7204 2067 9
Library of Congress Catalogue Card Number 12-88575
PRINTED IN POLAND
(D.R.P.)
FOREWORD
Those who begin to study mathematics usually find it difficult to
develop the habit of strictly formulating the ideas to be expressed, to
learn the methods of correct reasoning, and to comprehend the basic
concepts of mathematics. These difficulties seem to be caused, first,
by the lack of adequate training in mathematical logic, that is the
discipline whose tasks is to study deductive reasoning employed in
proving mathematical theorems; second, by the ignorance of the basic
concepts and methods used in set theory, now commonly applied in
all branches of mathematics and serving as the basis for introducing
and explaining fundamental mathematical concepts (relations, mappings, etc.); third, by the ignorance of the basic concepts of abstract
algebra, a discipline which has been developing vigorously and is now
affecting all the remaining branches of mathematics.
The present book covers those elements of mathematical logic,
set theory, and abstract algebra which will enable the reader to study
modern mathematics, which explains the title. The book has developed
from my lectures on Introduction to mathematics, which I have given
in Warsaw University for a couple of years, and is intended mainly
as a freshmen course in mathematics. Its scope goes much beyond
the Introduction to mathematics as formulated in the curriculum, and
this is why it may prove useful to other readers as well—those studying engineering, natural science, and the humanities—who want to
prepare for advanced mathematical studies or to become familiar with
elements of mathematical logic, set theory, and the basic concepts
of abstract algebra. The present author's intention was to make the
book form a complete whole and to encourage at least some readers
to a further study of those branches of mathematics.
The exposition of the subject matter, as given in the present book,
differs from the traditional approach. First of all, set theory is not
preceded by elements of mathematical logic. The present author's
VI
FOREWORD
experience has shown that a beginner finds set theory easier than
mathematical logic. Moreover, if the concepts of logic are applied to
set theory at a too early stage, the reader becomes accustomed to
a purely mechanical formulation of proofs and does not develop
mathematical intuition in grasping the concepts and theorems of set
theory. A further argument in favour of the order of presentation
of the subject matter is that the best way of demonstrating the application of the concepts of logic in defining mathematical concepts and in
proving theorems is t o do this when the student already has acquired
some knowledge of advanced mathematics. In view of the above the
present author has decided to discuss the elements of set theory before
the elements of logic. But, to make the exposition clearer and to accustom the reader to logical symbolism, the author introduces that
symbolism gradually, beginning with the first sections, and applies it
systematically. The elements of mathematical logic are presented mainly
from the point of view of their applications in mathematics, chiefly
in proving theorems. This is why a formalization of logic is avoided
and more attention is paid to the rules of inference than to the laws
of logic. This applies in particular to the propositional calculus.
The book consists of 14 chapters. Chapters I and III to XI cover
the elements of set theory. In view of the elementary character of the
book an intuitive concept of function (mapping, transformation) is
first introduced in Chapter III, and a precise definition of a function
is to be found only in Chapter V. The material has been selected from
the point of view of its usefulness in the study of other branches of
mathematics. This is why the arithmetic of cardinals and ordinals has
been completely disregarded. A terminological change has been introduced after Bourbaki: partially ordered sets are termed ordered
sets. Chapter II is concerned with mathematical induction and proofs
by induction, Chapter XII and XIII cover the elements of mathematical
logic. The propositional calculus, because of its elementary character,
is treated much more comprehensively than is the functional calculus.
The last section of Chapter XII illustrates a formal approach to the
propositional calculus and includes a simple proof of the completeness
theorem. The functional calculus is not presented as a formal system,
and its exposition is confined to those laws and rules of inference which
FOREWORD
vii
are most frequently used in mathematical reasoning, and to many
examples that point t o its applications in mathematics. Chapter XIV
is essentially only a supplement t o the book and explains the basic
concepts of abstract algebra, such as subalgebra, homomorphism, isomorphism, congruence, etc.
When working on this b o o k the author has made use of her lecture
notes Introduction to Mathematical Logic and Set Theory (in Polish),
covering her lectures at a Course in Applications of Mathematics,
organized in 1965 by the Polish Academy of Sciences, Institute of
Mathematics. These lecture notes have since been published by the
Polish Academy of Sciences Division for Training Research Staff.
In writing the part of the b o o k concerned with set theory the author
has drawn largely from Professor K. Kuratowski's Introduction to
Set Theory and Topology and from Set Theory by Professors A. Mostowski and K. Kuratowski (both now available in English-language
versions).
The author is indebted t o Professors S. H a r t m a n and A. Mostowski
for their suggestions about the manuscript version of the present b o o k ;
their criticism has helped her t o improve the original text and to remove
its various shortcomings.
Thanks are also due to Professor M . Stark for his suggestion that
the present book be written and published.
HELENA RASIOWA
Errata
page,
line
V7
319
28 6
59 4
9710
1562
215 ÎS
235,
287«
An-(An
for
read
tasks
shows
axiom
task
show
axioms
An-(Ai u . . . u An-i)
on the set
the ordinal
(α Λ ~ ß) => a
χ is
algebra
KJ...KJ
An-i)
in the set
an ordinal
(α A ~ ß)a =>
χ is a
algebras
H. Rastows, Intraduaipn to Modern Mathematics
CHAPTER I
T H E ALGEBRA O F S E T S
1. The concept of set
The concept of set is one of the fundamental concepts of mathematics.
As examples of sets one may q u o t e : the set of all books in a given
library; the set of all letters of the Greek alphabet; the set of all integers;
the set of all sides of a given polygon; the set of all circles on a given
plane.
The branch of mathematics concerned with the study of the general
properties of sets, regardless of the nature of the objects which form
1
those sets, is termed set theory ) and is regarded as the foundation
of modern mathematics. This discipline was founded by Georg Cantor
in the years 1871-1883.
The objects which belong to a given set are called its elements.
The statement that an element a belongs to a set A (or: that a is an
element of a set A) is written:
2
(1)
aeA ),
while a e A, b e A, c e A will often be abbreviated into: a, b, c e A.
The statement that a does not belong to a set A (i.e., a is not an
element of the set A) will be written
(2)
αφ A
or
~(aeA).
3
The symbol ~ will always stand for not or it is not the case that )
It is convenient to introduce in mathematics the concept of empty
set, i.e., the set which has n o elements. It may be said, for instance,
1
) Special reference should be made to the role of Polish mathematicians in developing this discipline, especially to the numerous papers by W. Sierpinski.
2
) The symbol e was introduced by G. Peano; it is the first letter of the Greek
word εστί (is).
3
) The symbol ~ is a distorted letter Ν (from nego, Lat. I deny).
2
I. ALGEBRA OF SETS
2
that "the set of all real roots of the equation x + 1 = 0 is empty"
instead of "there does not exist any real number which is a root of
2
the equation x + 1 = 0".
The empty set is denoted by O.
The set whose all elements are a1, ..., an will be denoted by
(3)
{al9...,an}.
A set may also consist of one element. F o r instance, the set of all
even prime numbers has exactly one element, namely the number 2.
A set whose only element is a will, by analogy to (3), be denoted by
(4)
{a}.
If every element of a set A is an element of a set B, then we say that
A is a subset of Β
We also say that the set A is contained in the set Β
or that Β contains A, which is written
A c Β
or
Β ^ A.
Fig. 1
The symbol c is called the symbol of inclusion. By definition, A c Β
if and only if the following condition is satisfied for every x: if χ e A,
2
Hereafter, the words if... then... will often be replaced
then xeB ).
by the symbol =>, and the words if and only if by the symbol o .
Accordingly the above formulation may be written in symbols a s :
(5)
(A a B)o
1
(for every χ: χ e A => χ e B).
) If at the same time the sets A, Β are not identical (cf. p. 3), it is said that A
is a proper subset of B.
2
) The phrase: α if and only if β (where α and β are any formulas that have the
form of statements) means : if a, then β, and if β, then a.
1. CONCEPT OF SET
3
Examples
The set of all integers is contained in the set of all rational numbers,
since every integer is a rational number. The set A = { 1 , 2} is contained
in the set Β = { 1 , 2 , 3 } , since 1 eB and 2eB.
Let C and D be, respectively, areas of the circles shown in Fig. 1. The set C is a subset of the
set D9 since every element of the set C is an element of the set D. The
set of all irrational numbers is contained in the set of all real numbers,
since every irrational number is a real number.
The statement that A is not a subset of Β is written
Α φ Β
or
Β φ A.
The following notations are also used:
~(A
cz B)
or
~(B
ZD
A).
It follows from the definition of a subset that Α φ Β if and only
if not every element of the set A is an element of the set B9 that is, there
exists in the set A an element which is not an element of B. In symbolic
notation:
(6)
~(A
(there is an χ such t h a t : χ e A and
ci B)o
e B)).
Examples
Figs. 2-4 shows examples of sets A and Β such that ~(A
The sets A and Β are represented by areas of circles.
Fig. 2
Fig. 3
c
B).
Fig. 4
The set of all integers divisible by 3 is not contained in the set of
all integers divisible by 6, since there exists an integer divisible by 3
which is not divisible by 6, e.g., the number 9. The number 9 belongs
to the former set, but not t o the latter.
The sets A and Β are identical if and only if they have the same elements. This is written as follows:
(7)
{A = B) <=> (for every χ: χ e Α ο χ e B).
4
I. ALGEBRA OF SETS
Example
Let A be the set of all integers that are divisible by both 2 and 3,
and let Β be the set of all integers that are divisible by 6. The sets A
and Β are identical, since an integer is divisible by both 2 and 3 if and
only if it is divisible by 6, and hence the sets A and Β have the same
elements.
It follows from the definition of a subset that
1.1. For any sets A, B, C:
(8)
OŒA,
(9)
A cz
A,
(10)
if A cz Β
and Β cz C, then A cz C,
(11)
if A cz Β
and Β cz A , then A = By
(12)
if Αφ
B, then Α φ Β
or
Β φ A ,
Formula (8) states that the empty set is contained in every set. Since
the empty set does not have any elements, the condition that every
element of the set Ο is an element of a set A is satisfied
Formula (9) states that every set is a subset of itself. In fact, every
element of a set A is an element of A.
Formula (10) is the law of transitivity for the relation of inclusiont
To prove it let us assume that A cz Β and Β cz C. Then every elemen.
of the set A is an element of the set B, and every element of the set Β
is an element of the set C. Hence it follows that every element of the
set A is an element of the set C, and hence A cz C.
T o prove (11) let us assume that A cz Β and Β cz A. Hence every
element of the set A is an element of the set B, and every element of
the set Β is an element of the set A. Thus the sets A and Β have the
same elements, that is, they are identical.
Formula (12) follows from formula (11). Should it be that A cz Β
and Β cz A, then by (11) the sets A and Β would be identical, contrary
to the assumption that Α φ Β.
Formula (11) is often used in proving the identity of sets.
l
) Cf. example 1 in Chapter XIII, Section 1, p. 236.
2. UNION OF SETS
5
2. The union of sets
By the union (or the set-theoretic sum) of sets A and Β is meant the
set whose elements are all the elements of the set A and all the elements
of the set Β and which does not have any other elements. The union of
sets A and Β is denoted by A uB. It follows from the definition of the
union of sets that χ e Α ν Β if and only if χ is an element of at least one
of the sets A, B, that is, if and only if χ e A or χ eB. The word or will
often be replaced by the symbol ν . When this convention is adopted,
then the necessary and sufficient condition for χ e A uB is written in
symbols a s :
(1)
(x e A uB) ο (χ e Α ν χ e Β).
Examples
The set of all real numbers is the union of the set of all rational
numbers and the set of all irrational numbers. The set of all rational
numbers is the union of the set of all integers and the set of all rational
numbers. If A = {1, 2} and Β = {2, 3 } , then A KJB = { 1 , 2 , 3 } . One
more example is illustrated by Fig. 5, where the sets A and Β are represented by areas of circles, and the set A uB consists of the points
belonging to the shaded area.
Fig. 5
Consider now the case in which χ is not an element of the union
AKJB. By definition, χφΑνΒ
if and only if the condition that χ is
an element of at least one of the sets A , Β is not satisfied, that is, if
and only if χ φ A and χ φ Β. The word and will often be replaced by
the symbol Λ . Consequently, the necessary and sufficient condition
for χ φ A uB is written in symbols a s :
(2)
(χ φ A u2?) <=> (χ φ Α ΛΧ φ Β).
6
I. ALGEBRA OF SETS
The next theorem*) follows from the definition of the union of
sets:
2.1. For any sets A, B9 C:
(3)
(4)
AvB
=
Au(BuC)
BvA,
= (AUB)KJC
(5)
OuA=A,
(6)
AvA=A.
),
2
T o prove the formulas given above it is demonstrated that for every
χ the following condition is satisfied: χ is an element of the set on the
one side of a given equation if and only if it is an element of the set
on the other side of the equation. Thus, χ eB KJA if and only if χ belongs
to at least one of the sets Α , B, but this means that χ e Α ν Β and proves
(3). Equation (4) follows from the fact that χ eA\J{B\JC)
if and
only if χ is an element of at least one of the sets A , B9 C; likewise,
χ e (A \JB)KJC if and only if χ belongs t o at least one of the sets A ,
B, C. Since Ο is the empty set, hence χ e Ο ν A if and only if χ e A ,
which proves (5). Finally, χ e A ν A if and only if χ e A , which proves (6).
Equations (3) and (4) are, respectively, the law of
commutativity
and the law of associativity for the union operation on sets. They are
analogous in form to the respective arithmetical laws for the addition
of real numbers. Likewise, equation (5) is analogous to a similar law
valid in arithmetic. On the other hand, equation (6), termed the law
of idempotence for the union operation on sets, has n o analogue in
arithmetic.
The following theorem expresses the relationships between inclusion
and union of sets.
2.2. For any sets A , B, C, D :
(7)
A <= A u J ? ,
(8)
BCZAKJB,
*) The theorems given in Sections 2, 3, 4, 5 are due to the English mathematician G. Boole (1813-1864), whose studies initiated research in mathematical logic.
2
) Law (4) makes it possible, when adding a finite number of sets, to omit the
parentheses indicating the order of operations.
3. INTERSECTION OF SETS
7
(9)
if A cz C and Β cz C, then AvB
(10)
if A cz Β and C cz D, then A u C c
OD
A cz Β
if and only if
cz C,
BvD,
A uB = B.
Formulas (7) and (8), which follow directly from the definitions
of inclusion and union of sets, state that the union of sets contains
each of the summands.
T o prove (9) assume that A cz C and Β cz C. If χ e A vB, then χ
belongs t o at least one of the sets A, B. Suppose that χ e A holds. Since
A cz C, hence χ e C. By assuming in turn that χ e B, it can be inferred,
on the strength of the inclusion Β cz C, that χ e C holds too. Hence,
if χ e A vB, then χ e C, which shows t h a t A vB CZ C and proves (9).
Formula (9) states that any set which contains two given sets also
contains the union of those sets.
T o prove (10) let us assume that A cz Β and C cz D. N o w , (7) and
(8) yield Β cz Β u D and D cz Β uZ>. Hence, by inclusions stated earlier,
A cz BvD and C cz BvD follows from 1.1 (10). Consequently, on the
strength of (9) we may conclude that AKJC cz BvD, which completes
the proof of (10).
Formula (11) makes it possible to define the concept of inclusion of
sets by the union and the identity of sets. T o prove (11) let us assume
that A cz B. Since at the same time Β cz B, (9) yields A vB cz B. But
Β cz A vB by (8). The last two conclusions, by 1.1 (11), yield Α ν Β = Β.
Assume now that A vB = B. This and (7) yield A a B, which completes
the proof of (11).
3. The intersection of sets. The laws of absorption and distributivity
By the intersection (or the set-theoretic product) of sets A and Β is
meant the common part of those sets, that is the set which includes those
and only those elements which belong both to the set A and to the set
B. The intersection of sets A and Β will be denoted by A nB. It follows from the definition of the intersection of sets that χ e A nB if and
only if χ eA and X e B, or, in symbolic notation,
(1)
(x e Α η Β) ο (χ e Α Λ Χ e Β).
8
I. ALGEBRA OF SETS
Examples
The set of all integers divisible by 6 is the intersection of the set of
all even integers and the set of all integers divisible by 3. The empty
set is the intersection of the set of all rational numbers and the set of
all irrational numbers. If A = {1, 2} and Β = {2, 3}, then AnB = {2}.
Another example is illustrated by Fig. 6, where the sets A and Β are
represented by areas of circles, and the set A nB consists of the points
belonging to the shaded area.
Fig. 6
It follows from the definition of the intersection of sets that χ φ A nB
if and only if χ does not belong to both A and B, that is, χ φ A or χ φ Β,
or, in symbolic notation,
(2)
(χφΑηΒ)ο(χφΑνχφΒ).
It follows from the definition of the intersection of sets that
3.1. For any sets A, B, C:
(3)
(4)
(5)
(6)
A nB =
An{BnC)
OnA
BnA,
=
=
(AnB)nC
0,
AnA=A.
The definition of the intersection of sets states that x e A nB if and
only if x e A and x e B, that is, x e Β and χ e A9 and that means that
χ e BnA. This proves formula (3).
Formula (4) follows from the fact that x e A n(BnC)
if and only if
χ belongs to each of the sets A, B, C; likewise, χ e (A nB)nC
if and
only if χ belongs to each of the sets A, B, C.
*) Law (4) makes it possible, when multiplying a finite number of sets, to omit
the parentheses indicating the order of operations.
3. INTERSECTION OF SETS
9
The set Ο nA is empty, for should it be not empty, it would have at
least one element x, which by (1) would yield χ eO and χ e A, which
is not possible, since Ο is the empty set. This proves (5).
Formula (6) follows directly from the definition of the intersection
of sets, since χ e A η A if and only if Λ: G Α.
Formulas (3) and (4) are, respectively, the law of commutativity and
the law of associativity for the intersection operation on sets. They are
analogous in form to the respective arithmetic laws for the multiplication of real numbers. Formula (5), too, is analogous t o the respective
arithmetical law. On the contrary, formula (6), termed the law of idempotence for the intersection operation on sets, has n o analogue in
arithmetic.
The following theorem describes the relationships between inclusion
and intersection of sets.
3.2. For any sets A, B, C, D:
(7)
AnB
cz A,
(8)
AnB
cz B,
(9)
if A cz Β and A cz C, then A cz
(10)
OD
if A cz Β and C cz D9 then AnC
BnC,
cz
BnD,
A cz Β if and only if A nB = A.
Formulas (7) and (8) follow directly from the definitions of inclusion
and intersection of sets. They state that the intersection of sets is contained in each of its factors.
T o prove (9) let us assume that A cz Β and A cz C. If χ G A, then
by these assumptions xeB
and χ e C, and hence xeBnC.
Consequently, A cz Br\C. Formula (9) states that any set contained in two
given sets is also contained in the intersection of those sets.
T o prove (10) assume that A α Β and C cz D. N o w (7) and (8)
yield A n C cz A and A n C cz C. This and the inclusions mentioned
earlier yield, by 1.1 (10), A nC cz Β and A nC cz D. On applying (9)
it can be concluded that A n C cz BnD9 which completes the proof of
(10).
Formula (11) makes it possible t o define the concept of inclusion of
sets by the concepts of intersection and identity of sets. T o prove (11)
I. ALGEBRA OF SETS
10
let it be assumed that A c B. Since A c A, then by (9) A cz A nB.
Hence, and from (7)— on the strength of 1.1 (10)—it can be concluded
that A nB = A. Assume now that A nB = A. On the strength of (8)
it can be concluded that A = A nB c B, that is, A cz B. This completes the proof of (11).
Two sets, A and B, are said t o be disjoint if A nB = 0 , that is, if
they do not have any element in common. This term will often be used
in sections that follow.
Examples
The set of all even integers and the set of all odd integers are disjoint sets. The set of all non-negative real numbers and the set of all
negative real numbers are disjoint sets. The set A = { 1 , 2 } and the set
Β = { 3 , 4 } are disjoint sets.
The laws linking the operation of union of sets with that of intersection of sets will be given below.
3.3. For any sets A, B, C:
(12)
An(AuB)
= A,
(13)
(AnB)uB
= B,
(14)
An(BuC)=
(15)
Au(BnC)=
(AnB)u(AnC),
(AKJB)n(A
uC).
By 3.2 (11), formula (12) is equivalent to the inclusion A a A KJB,
which is always satisfied (cf. 2.2 (7)).
Likewise, formula (13) is, by 2.2 (11), equivalent t o the inclusion
A nB cz B, which is always satisfied for any sets A9 B (cf. 3.2 (8)).
Formulas (12) and (13) are thus proved. They are termed the laws of
absorption. N o t e that the analogous laws do not hold in the arithmetic
of real numbers.
Formula (14) is the law of distributivity of intersection over union and
is an analogue of the arithmetical law of distributivity for multiplication
over addition of reals. T o prove (14) assume that x e A n(BvC).
Thus
x e A and χ e BKJC. Hence x e A and χ is an element of at least one
of the sets B, C. If x e B, then, since x e A, x e A nB. Consequently
χ e (A nB) v(AnC),
since χ is an element of one of the summands of
4. DIFFERENCE OF SETS
11
this union. If χ e C , then, since χ e A, χ e AnC. Hence it follows that
xe(A nB) \j(Ar\C).
Conversely, assume that χ e (A nB)u(A n C ) . It
follows that χ is an element of at least one summand of this union. If
and hence
χ e A nB9 then χ e A and χ e B. Thus χ e A and χ eBvC,
χ eAn(BvC).
If xeAnC,
then xeA
and xeC.
Consequently,
χ G A n ( 5 u C ) . The proof of (14) is thus completed.
Formula (15) is the law of distributivity of union over intersection. It
is to be noted that an analogous law does not hold in the arithmetic
of real numbers. Formula (15) can be proved by analogy t o the proof
of (14). A different proof will be given here, namely one which refers
t o equations (14), (12) and (13), as well as t o commutativity and
associativity of union and intersection of sets. O n the strength of the
laws referred to above the following equations hold:
(A u i ) n(A u C ) = [(A u 5 ) nA] u[(A u i ) n C ) ]
=
[An(AyjB)]u[(AuB)nC]
=
Au[(AnC)u(BnQ]
=
[Au(AnC)]u(BnC)
= [(CnA)uA]Kj(BnC)
=
Au(BnC)9
which prove (15).
4. The difference of sets. Relationships between the difference of sets and
the operations of union and intersection of sets
The set consisting of those and only those elements which belong to
a set A and d o not belong t o a set Β is termed the difference of the sets
A and B. The difference of the sets A and Β is denoted by A — B. It
follows from the definition of the difference of sets that
(1)
(x eA-B)o
(x eA ΑΧ φ Β).
Examples
The set of all irrational numbers is the difference of the set of all
real numbers and the set of all rational numbers. If A is the set of all
natural numbers (i.e., positive integers) and Β is the set of all integers,
then the difference A— Β is the empty set, since there is no natural
number which is not an integer. If A = { 1 , 2 , 3 } , Β = { 2 , 3 , 4 } , then
12
I. ALGEBRA OF SETS
A—Β = {1}. The next example is illustrated by Fig. 7, where the sets
A and Β are represented by areas of circles. The set A-Β consists of
the points belonging to the shaded area.
Fig. 7
It follows from the definition of difference of sets that χ φ A—Β if
and only if the conditions x e A and χ φ Β are not both satisfied. This
occurs if and only if at least one of these conditions is not satisfied,
that is, χ φ A or x e B. In symbolic notation:
(2)
ο (χ φ Α ν χ Ε Β).
(χ φ Α-Β)
The following theorem lays down the relationships between inclusion
and difference operation on sets.
4.1. For any sets A, B, C, D:
(3)
Α-B
cz A,
(4)
if A cz Β and C cz D, then Α-D
cz
(5)
if C cz D, then Α-D
A-C,
(6)
A cz Β if and only if Α-Β
cz
B-C,
= Ο.
Formula (3) follows directly from the definitions of inclusion and
set difference. For if x e A — B, then x e A and χ φ Β. Hence x e A, and
thus A—Β cz A.
Suppose that the assumptions of (4) are satisfied, so that A cz Β and
C cz D. If x e A — D, then x e A and χ φ D. Since A cz Β, the condition xeA
implies xeB.
Since C cz D, the condition χφΌ
implies
χ φ C, for should χ be an element of C then it would be an element
of D. Consequently we may conclude that if χ e A — D, then x e Β and
χ φ C, and hence x e B—C. The proof of (4) is thus complete.
Formula (5) follows from 1.1 (9) and 4.1 (4), where A is substituted
for B.
4. DIFFERENCE OF SETS
13
T o prove (6) note that the condition A-B = Ο means that the
set A does n o t include any element χ that is n o t included in the set B,
so that every element of A is an element of B, and this in turn means
that A a B.
The following theorem lays down the relationships between set difference on the one hand a n d set union a n d set intersection on the other.
4.2. For any sets A, B, C, D:
(7)
A-(BuC)
=
(A-B)n(A-C)9
(8)
A-(BnC)
=
(A-B)KJ(A-C).
1
Formulas (7) a n d (8) are termed De Morgan's laws ) (for set difference). T o prove (7) note that χ e A-(BKJC)
if and only if xeA
and χ φBKJC.
But χ does n o t belong t o the union BuC if and only
if χ does n o t belong t o any summand of that union, that is, χ φ Β a n d
χ φ C . The condition χ e A a n d χ φ Β a n d χ φ C is equivalent t o the
condition xeA-B
a n d χ eA-C,
which in turn is equivalent t o the
condition χ e (A-B)n(A-C).
This demonstrates that χ e
A-(BvC)
if and only if χ e (A - B) n(A - C ) , which proves (7).
F o r m u l a (8) is proved analogously. T h e condition χ e A-(BnC)
is
equivalent t o the condition χ s A a n d χ φΒηϋ.
But χ φ BnC if and
only if χ does n o t belong t o at least one of the factors. The condition
χ e A a n d χ does n o t belong t o at least one of the sets B, C is in turn
equivalent t o the condition that χ belongs t o at least one of the sets
that is, the condition χ e (A — B)v(A-C).
This demonΑ-B,
A-C9
strates that χ e A - (Β η C) if and only if χ e (A - B) u (A - C), which
proves (8).
F o r any sets A, Β the sets A and B — A are disjoint, so that A n(B—
= O. F o r should χ e A n(B-A),
-A)
then χ e A and χ e B-Α
hold, which would imply that χ G A a n d xeB
would
and χ φ A, which is im-
possible.
The following theorem refers to the union of the sets A a n d B—A.
4.3. For any sets A, B:
(9)
Au(B-A)
= Au5,
(10)
if A cz B9 then A v(B-A)
= B.
J
) After A. De Morgan (1806-1871), an English logician.
14
I. ALGEBRA OF SETS
Assume that χ e A vB. Hence χ belongs to at least one of the sets
A, B. If χ e A, then obviously χ e A u(B-A).
If χ φ A, then χ e B.
Then both χ φ A and χ e B, so that χ eB-A.
Consequently, χ
eAu
u(B-A).
This demonstrates that Α ν B c A u(B-A).
By 4.1 (3), B-A cz B. Since A cz A, by 2.2 (10) we may conclude that A
v(B—A)czA u i ? . This and the inclusion proved above yield (9).
If A cz B, then AuB = B (cf. 2.2 (11)). F r o m this and from (9) it
can be concluded that A \J(B-A)
= B, which proves (10).
The following theorem makes it possible to define set intersection
by means of set difference.
4.4. For any sets A, B:
(11)
A-(A-B)
==
AnB.
By (2), χ φ A—Β if and only if at least one of the following conditions is satisfied: χ φ A, xeB.
Hence, if xeA — {A — B), then
xeA
and χ also satisfies at least one of the conditions: χ φ A, χ e Β. Since
the conditions χ e A and χ φ A cannot both be satisfied, the condition
χ e Β must be satisfied. Consequently, xeA and χ e B, so that χ
eAn
nB. This proves that A-(A-B)
czAnB.
If xeAnB,
then
xeA
and χ e Β. F r o m this we infer that xeA and χ φ A — Β, so that χ s A —
-(A-Β).
This proves that A nB cz A-(A-B).
The proof of 4.4 is
thus completed.
The following equation will be proved :
4.5. For any sets A, B, C:
(12)
A-(BuC)
=
(A-B)-C.
The proof of Theorem 4.2 (7) shows that χ e A-(BvC)
if and only
and χ φ C , which is satif χ e A, χ φ B9 and χ φ C, that is, xeA-Β
isfied if and only if χ e (A—B) — C. This completes the proof of Theorem 4.5.
5. The universe. The complement of a set
The applications of the algebra of sets are usually confined to analyses of sets which are subsets of a certain fixed set termed a space or
a universe. For instance, in arithmetic the universe is the set of natural
5. UNIVERSE. COMPLEMENT OF A SET
15
numbers; in analysis, the set of real n u m b e r s ; in geometry, the set of
points in the Euclidean space.
When the subsets of a given universe X are considered, the concept
of subset is often identified with that of the property which is an attribute of every element of that subset and is not an attribute of any
other element of that universe. Then, if A cz Χ, χ e A is replaced in
writing by A(x) and read χ has the property A. F o r instance, if the
universe is the set Jf of all natural numbers and if A is the set of all
natural even numbers, then x e A is replaced by A(x) which is interpreted a s : Λ: is an even number. Likewise, if Ρ is the set of all natural
prime numbers, x e Ρ may by written P(x) and interpreted as : χ is
a natural prime number.
The subsets of a fixed universe X are also called singulary (or unary)
relations in X.
Let X stand for a fixed universe. The set X— A is called the complement of the set A cz X and is denoted by —A. By definition, —A is
the set of all those elements of the universe X which are not elements
of the set A. It is t o be noted that the complement of a set depends
on the choice of the universe. F o r instance, in the universe of natural
numbers the set A = { 1 , 2 , 3 } has as its complement the set of all those
natural numbers which are greater than 3 ; but in the universe X = { 1 ,
2 , 3 , 4 , 5} the complement of A is the set {4, 5 } .
If the universe X is fixed, then it follows from the definition of set
complement that for any A cz X and for any x e X the condition
(1)
(χ e -Α)
ο (χ φ A).
It follows from the definitions of universe and set complement that
5.1. For any subsets A, Β of a universe X:
(2)
XrsA
= A,
(3)
XuA
= X,
(4)
-X
= o,
(5)
-O
= X,
(6)
A = A,
(7)
A <= Β if and only if — B cz —A.
16
I. ALGEBRA OF SETS
Formula (2) follows from the assumption that A cz X and from
3.2(11).
Likewise, by 2.2 (11), formula (3) is just a different formulation of
the statement that A cz X.
The complement of the set X is the empty set, since the conditions
x e X and χ φ X cannot both be satisfied.
The complement of the empty set is the whole universe X, since — Ο
contains every element of the set X which is not an element of the empty
set, that is, every element of the set X.
T o prove (6) note that an element χ of the set X belongs to
A
if and only if χ φ — A. The last condition is equivalent to χ e A, for if
χ φ — A, then χ must belong to A9 since otherwise it would be an element
of — A, which is contrary to the assumption made. Hence χ Ε A. Conversely, if ΛΓ Ε A, then χ φ —A9 since otherwise condition (1) would not
be satisfied. F o r instance, let X = { 1 , 2 , 3 , 4 , 5} and let A = { 1 , 2 } .
Then -A = { 3 , 4 , 5 } , - - A = - { 3 , 4 , 5 } = { 1 , 2 } = A.
Formula (7) follows directly, by 4.1 (5), from the following formulas:
-B = X-B9
and (6).
-A = X-A9
The following theorem expresses the relationships between any set
and its complement.
5.2. For any subset A of a universe X:
(8)
Au-A
=
X,
(9)
An-A
=
0,
(10)
-(An-A)
= X,
(11)
-(Au-A)
= O.
If χ EX, then χ Ε A or χ φ A, so that χ belongs to at least one of
the sets A and —A, hence χ E Au —A. If χ E Au— A9 then x belongs
to at least one of the sets A and — A, which are subsets of X9 and hence
χ EX. This proves (8), which states that the union of any subset A of
a universe X and its complement equals the whole universe X.
Formula (9) follows from the fact that should χ Ε An— A, then the
conditions χ Ε A and χ φ A would both have to be satisfied, which is
impossible. Hence there does not exist any element that would belong
t o An—A,
which means that An —A = O. Formula (9) states that
every set is disjoint from its complement.
17
5. UNIVERSE. COMPLEMENT OF A SET
Formula (10) follows directly from (9) and 5.1 (5).
Formula (11) follows directly from (8) and 5.1 (4).
The following theorem results from Theorem 4.2 a n d the definition
of set complement.
5.3. For any subsets A, Β of a universe X:
(12)
-(AvB)
= — An—B,
(13)
-(AnB)
=
-Au-B.
Formulas (12) a n d (13) are De Morgan's laws. They state that the
complement of the union of two subsets of a universe X equals the
intersection of their complements, a n d that the complement of the intersection of two subsets of a universe X equals the union of their complements. T o prove these laws we need merely substitute, in equations
4.2 (7), (8), the universe X for A, t h e set A for B, and the set Β for C
and use the definition of set complement as the difference of a universe
X and a given set.
The theorem that follows lays down the relationships between the
difference of any subsets of a fixed universe a n d the operations of set
complementation, set union, and set intersection.
5.4. For any subsets A, Β of a universe X:
(14)
A-B
=
An-B,
(15)
A-B
= —( — A uB).
T o prove (14) note that χ eA — B if and only if x e A a n d χ φ B,
that is, x e A and x e — B, which means that xeAn
— B.
Equation (15) results from (14), (6), and (12), since these yield
A-B
= An—B
=
A n—B
=
—(-A
KJB).
The following theorem should also be noted.
5.5. For any subsets A, Β of a universe X:
(16)
A c Β if and only if Α η —Β = Ο,
(17)
A c Β if and only if -A \JB = X.
Equivalence (16) results from 4.1 (6) and 5.4 (14).
T o prove (17) we show that the condition An—Β = Ο is equivalent
t o the condition -AKJB
= X. If A n - B = O, then, by 5.1 (5), -
(An
18
I. ALGEBRA OF SETS
n-B)
= X. But, by 5.3(13) and 5.1(6), -(An-B)
=
-AKJ--B
= —AKJB.
Thus the condition An—B = 0 implies the condition
-AuB
= X. Conversely, if -AuB = X, then, by 5.1 (4),
-(-AKJ
KJB) = 0. But, by 5.3(12) and 5.1(6), ~(-AuB)=
A n—B
= An—B.
Thus the condition — AKJB = X implies the condition
An—Β = Ο. Hence these two conditions are equivalent. This fact,
together with (16), yields (17).
6. Axioms of the algebra of sets
Those properties of sets which have been referred t o so far can be
formulated as a system of axioms which yield all the theorems of the
algebra of sets given in Sections 1-5.
The concept of set and the concept of elementhood (x e A) are
adopted as primitive concepts. The following four axioms form a system
of axioms of the algebra of sets.
(I) T H E AXIOM OF EXTENSIONALITY OF SETS. If the sets A and Β have
the same elements, the sets A and Β are identical.
(II) T H E AXIOM OF UNION. For any sets A and Β there exists a set
whose elements are precisely all the elements of the set A and all the
elements of the set B.
(III) T H E AXIOM OF DIFFERENCE. For any sets A and Β there exists a set
whose elements are precisely those elements of the set A which are not
elements of the set B.
(IV) T H E AXIOM OF EXISTENCE. There exists at least one set.
Set inclusion can be defined by means of the union of sets and the
identity of sets (2.2(11)). The intersection of sets can be defined by
means of set difference (4.4(11)). The empty set can be defined as
A— A for any set A (the existence of at least one set follows from
Axiom (IV)).
All the theorems given so far can be deduced from these axioms. In
the proofs reference is made t o those properties of sets only which are
formulated in the axioms, a n d not t o any intuitions connected with the
concept of set.
7. FIELDS OF SETS
19
7. Fields of sets
In various mathematical considerations reference is often made to
sets whose elements are sets. Such sets will be said to be families of sets.
Let 91 be a family of subsets of a fixed universe Χ Φ Ο. The family 91
will be termed a field of sets if the following conditions are satisfied:
(c x ) 91 is non-empty,
(c 2 ) if A G 91, then -A
e9l,
(c 3 ) ifAe9iandBe9l,
then
AvBe9l.
7.1. If a family 91 of subsets of a universe Χ Φ Ο is a field of sets,
then Ο e9l andX
e9t.
It results from Condition (c x ) that at least one subset A of the universe Ζ belongs t o 91. It can be inferred from (c 2 ) that the set —A also
belongs to 91. By ( c 3 ) , the union of two sets that belong to 91 also
belongs t o 91, and hence the set A u — A e 91. But, by 5.2 (8), Au—A
= X. Hence Xe9i.
Since, by ( c 2 ) , the complement of every set that
belongs t o 91 also belongs to 91, then —X e 91. But —X=0
(cf. 5.1 (4)),
so that Ο e9t.
The next theorem states that in the definition of a field of sets Condition (c 3 ) can be replaced by the following: if the sets A and Β belong
t o 91, then their intersection also belongs t o 91.
7.2. A necessary and sufficient condition for a family 91 of subsets of
a universe Χ Φ Ο to be a field of sets is that the following conditions be
satisfied: ( c j , ( c 2 ) , and:
(cf) ifAe9landBe9l,
then AnB
e 91.
It will be proved first that Condition (c%) is satisfied in every field
of sets. By D e Morgan's law 5.3 (13), -(AnB)
= -Au-B.
Hence,
the application of 5.1 (6) yields
AnB
=
(A nB) =
-(-Au-B).
If the sets A and Β belong to the field of sets 91, then it follows from
and —iieÇR; hence the union of those
(c 2 ) and (c 3 ) that — Ae9l
sets —AKJ—BG91
and accordingly the complement of that set
— (—Au —B) = AnB e 91. Assume now triât Conditions ( c j , (c 2 ),
and (c?) are satisfied. We show that Condition (c 3 ) is also satisfied,
20
I. ALGEBRA OF SETS
so that 91 is a field of sets. By De Morgan's law 5.3 (12),
-(AKJB)
= —A n — B. This, by 5.1 (6), yields AuB =
(AuB) =
— ( — An—B). If the sets A and Β are in 9\, then, by ( c 2 ) , —A G 9V
and
Hence, by (c%), —A n — B e 9?. Then, by ( c 2 ) , it follows
that —( — An — B) = A u i ? e 9Î, which completes the proof.
We now list a number of equations concerning the empty set and
the universe which hold in every field of sets.
ΧΦ
7.3. The following equations hold in any field of subsets of a universe
O:
O u O = o,
0)
(2)
O u i = XuO
(3)
Xr\X
= X,
= XnO
= OnO
(4)
OnX
= XKJX
2.1 (6),
= X,
5.1 (3), 2.1 (3),
3.1 (6),
= o,
5.1 (2), 3.1 (3).
(5)
- 0
= X,
5.1 (5),
(6)
-X
= 0,
5.1 (4),
Examples
The family of all subsets of any universe Χ φ Ο is an example of
a field of sets. Another simple but important example of a field of
sets is that of a family whose only elements are the empty set Ο and
the whole universe X. This family is a field of sets on the strength of
7.3, since this theorem ensures that Conditions (c 2 ) and (c 3 ) are satisfied,
and Condition (c x ) is satisfied by the very definition of this family
of sets. Another example of a field of sets is provided by the family
of all finite subsets of a universe Χ Φ Ο and their complements.
8. Prepositional functions of one variable
Given a universe Χ Φ (9, a formula ψ(χ) in which the variable λ*
occurs and which becomes a true or a false proposition when the name
of any element of the universe X is substituted for the variable x, is
called a propositional function (or a predicate) of one variable x, which
ranges over the universe X. The set X is then called the range of .v.
The notation
(1)
ψ(χ),
XGX,
8. PROPOSITIONAL FUNCTIONS OF ONE VARIABLE
will be used to indicate that χ in φ(χ)
21
ranges over X.
Examples
Let 2t be the set of all integers. The formula
2
(2)
m -2m
> 0,
m e &,
is an example of a propositional function of the variable m which ranges
over
that is, a propositional function whose variable ranges over
the set of integers. Let 0t stand for the set of all real numbers. The
formula
2
Χ -3Λ: + 2 =
(3)
0,
Χ E
0t,
is an example of a propositional function of the variable χ ranging
over the set of all real numbers. Let # be the set of all complex numbers.
The formula
(4)
|z|<l,
Z G ^ ,
is an example of a propositional function of the variable z, ranging
over the set of all complex numbers.
Consider any propositional function φ(χ), χ Ε X. The element α Ε Χ
is said to satisfy the propositional function φ(χ), if the proposition φ(α)
is true. For instance, the integer 3 satisfies the propositional function (2),
2
since 3 — 2 · 3 > 0 . The real number 2 satisfies the propositional
2
function (3), since 2 — 3 - 2 + 2 = 0. Likewise, the complex number
(0, - 1 ) satisfies the propositional function (4), since |(0, —1)| < 1.
The set of all those values of the variable χ Ε X for which the p r o positional function φ(χ), XEX, becomes a true proposition, that is,
the set of those x's which satisfy that propositional function, is denoted by
(5)
{XEX:
φ(χ)}.
By definition of the set (5), an element α e A" is in the set (5) if and
only if a satisfies the propositional function φ(χ).
For every a EX
(6)
(a e {x e Χ: <p(x)}) ο
φ(α).
In symbols:
I. ALGEBRA OF SETS
22
Examples
The set
{meZ:
2
m -2m
> 0} = 3T-{09
1,2},
since every integer m > 2 satisfies the propositional function (2) and
every negative integer satisfies that propositional function, but the
numbers 0, 1, 2 do not satisfy that propositional function.
The set
{xe
2
9t\ x -3x
+ 2 = 0} = { 1 , 2 } ,
since the real numbers 1 and 2, and those real numbers only, satisfy
the propositional function (3).
The set { z e ^ : \z\ < 1} is the set of all those complex numbers
whose geometrical images are points on the Gaussian plane, situated
at a distance not greater than 1 from the origin of the system of coordinates, that is, points of a circle whose centre is at the origin of
the system of coordinates and whose radius equals 1.
Every propositional function <p(x), χ eX, thus determines a subset
{x G Χ: φ(χ)} of the universe X, that is, a certain property of elements
of the set X, in accordance with what has been said in Section 5.
9. Note on axioms of set theory
The axioms of the algebra of sets specified in Section 6 do not formulate all those fundamental properties of the concept of set which are
required in the present b o o k ; in particular, they do not suffice in the
discussion made in Section 7 and 8 of this chapter. The following
three axioms *) will be added to the axioms of the algebra of sets in
order to form a system of axioms that expresses the properties of the
2
concept of set as required both in the present book ) and in the ap*) Axioms (l)-(IV) (Section 6) and (V)-(VII) are due to the German mathematician E. Zermelo; they were published in his paper Untersuchungen über die
Axiome der Mengenlehre, Mathematische Annalen 65 (1908), pp. 261-281.
2
) The axiom system adopted here does not suffice to cover only the discussion
made in Chapter XIV, Sections 6-9.
10. ON THE NEED OF AN AXIOMATIC APPROACH TO SET THEORY
23
plications of set theory in other branches of mathematics. This system
will suffice if we assume the existence of the set of natural numbers
which satisfies Peano's axiom system (see Chapter II). The above system
of seven axioms together with the arithmetic of natural numbers does
not, however, suffice t o develop set theory in its most sophisticated
form.
( V ) T H E AXIOM OF SUBSETS. For every set X and for every
propositional
function φ(χ) ranging over X there exists a set consisting of precisely
those elements of the set X which satisfy that propositional function.
(VI) T H E AXIOM OF POWER SET. For every set X there exists a family
of sets whose elements
are precisely
the subsets of the set X. (It is
X
called the power set of the set X and denoted by 2 .)
(VII) T H E AXIOM OF CHOICE * ) . For every family
of non-empty
disjoint
sets there exists a set which has one and only one element common with
each set of the family.
Axioms (V) and (VI) have been resorted t o in Sections 7 and 8
above, while the axiom of choice has not been used so far. Reference
to it will be made in further chapters.
It is worth noting that by joining Axioms (V)-(VII) t o Axioms
(I)-(IV) of the algebra of sets one could at the same time drop some
of the previous ones. This applies, for instance, t o the Axiom of difference (III), since the existence of a set A — Β = {xeA: χ φ Β} for
any sets A, Β follows from the Axiom of subsets. In view of the elementary character of the present book the possibilities of eliminating other
axioms of the algebra of sets will n o t be discussed here.
10. Comments on the need of an axiomatic approach to set theory and on
axiomatic theories
In the early stage of its development, set theory was founded on
an intuitive concept of set. Cantor, the founder of set theory, did not
introduce any concept of set axiomatically, n o r did he give any de*) Compare the remarks pertaining to the axiom of choice, Chapter XI, Section 7.
I. ALGEBRA OF SETS
24
finition of a set. The concept was being used freely on the strength
of the intuitions associated with it. This method, however, proved
unreliable, since intuition did not provide unambiguous answers t o
more sophisticated questions about the properties of sets. This led t o
paradoxes in set theory, which could not be eliminated by reference
to intuition alone. The following Russell's paradox can serve as an
example *).
Consider a set Ζ that consists of all those sets which are not elements
of themselves. Thus a set A is an element of the set Ζ if and only if A
is not an element of A. The concept of the set Ζ results in a contradiction: consider now whether Ζ is an element of Z . If it is, then, by definition of the set Z , it follows that Ζ is not an element of itself, that
is, Ζ is not an element of the set Z . But if Ζ is not an element of itself,
then, by definition of the set Z , it follows that Ζ is an element of the
set Z . This yields a contradiction.
Russell's paradox could be constructed only because an inexact,
intuitive concept of set was being used.
As this example shows, in mathematics reference to intuition alone
may result in contradictions. N o w axiomatic theories impart precision
to intuitive mathematical theories. The necessity for constructing
mathematical theories on axiomatic principles was felt even in antiquity. The axiomatic system of geometry, as presented by Euclid (4th
cent. B.C.) in his Elements, has survived to the present times.
In an axiomatic theory, certain concepts, which are to figure in
it, are selected first and described by a system of axioms. Those concepts are termed the primitive concepts of that theory. In an axiomatic
theory, only primitive concepts, or concepts defined by reference to
primitive concepts, may be used. Only those statements which can be
deduced from axioms by correct reasoning are accepted as theorems.
All those properties of primitive concepts occurring in a given theory
which are not specified by the axioms must be proved.
The paradoxes of set theory can be eliminated in axiomatic set
l
) This paradox, discovered by Bertrand Russell, was published in 1903. A similar
paradox had been discovered even earlier hy the Italian mathematician C. BuraliForti.
EXERCISES
25
1
t h e o r y ) . The first formulation of a system of axioms for set theory
is due to E. Zermelo (1904). His system has since been modified many
2
t i m e s ) . At present there are various methods of axiomatizing
set
theory. Among these we mention that axiom system which includes
a concept more general than the concept of set, namely that of class.
This axiomatization of set theory is due to P. Bernays.
Exercises
1. Let Λ = {xe@: \x\ > 5},B=
{xe&: - 6 < χ < 0 } , where ^ stands for the
set of all real numbers. Interpret these sets diagrammatically. Determine the sets A vB,
AnBf —A,A—B, B—A, and interpret them diagrammatically.
2. Show that for any sets A and Β the following equations hold :
A-B
= A — (Ar\B),
A = (AnB)\j(A
— B).
3. Show that for any sets A, B, C the following equations hold:
An(B-C)
=
(AnB)-C,
(AuB)-C=
(A-Qv(B-C),
A-(B-C)=
(A-B)Kj(AnC).
4. Show by adequate examples that the equations
{A-B)VJB
= A,
(AvB)-B
= A
do not hold for any sets A and B.
5. Prove that, for any sets A and Bf if -Au-B
A = B. Is the converse theorem true?
= -A
and also Β <= A, then
6. Check whether the following propositions are true :
for any sets A, B:A a Bo
for any sets A, ByC:Aï
-An-Β
B=>C-A
=
φ
-B,
C-B.
7. For any finite set A, let N(A) stand for the number of its elements. Prove that
N{A uB) = N(A)+N(B)
Find a formula for Ν (A
- N(A η Β).
vBvQ.
8. The symmetric difference of the sets A, B, denoted by A—B, is defined as
A—Β = (A-B)u(B-A).
Prove that, for any sets A, Bt C, we have
x
) A comprehensive exposition of axiomatic set theory can be found for instance
in Set theory by A. Mostowski and K. Kuratowski, Amsterdam-Warszawa 1967.
2
) A. Fraenkel, J. v. Neumann, P. Bernays, K. Gödel, and others.
I. ALGEBRA OF SETS
26
(Α-Β
(a)
= Ο) ο
(A = Β),
Α-B
---= B-Α,
(c)
(A-B)-C
=
(d)
Λη(£ -C) =
(b)
A-(B-C),
(Ar\B)—(AnC),
Λ-^Ο = A.
(e)
Show that, for any sets A,B,
there exists exactly one set C such that Λ — C =
/////Λ C --= A—B.
9. For any subsets A, Β of a fixed universe .V, the union and the intersection of
those subsets can be defined by means of inclusion in the following way:
A yJB is the least subset of X containing the sets A and B,
AnB is the greatest subset of Xcontained in both A and B.
Establish the equivalence of these definitions with those given earlier in the main
text. Use the definitions given in this exercise to prove the formulas:
(A c: B)o(AvB
= B),
(AnB)nC
An(BnQ,
(AnB)vC=
=
(AvC)n(BvQ.
10. Prove that, for any non-empty family SR of subsets of a universe X, there
exists a least field & of subsets of X which contains SR, that is, which is contained
in any field of subsets of X containing SR. In particular, prove that the family 5 l 0 of
finite unions of finite intersections of sets belonging to SR, or of complements of sets
belonging to SR, i.e., sets of the form
(AnnA12n
... nAlni)v
... v(AmlnAm2n
... nAm„
),
where A-^ e SR or A\j == —Bij and B,j e SR, is the least field that contains SR.
CHAPTER II
NATURAL NUMBERS. PROOFS BY INDUCTION
1. An axiomatic approach to natural numbers. The principle of induction
The numbers 1, 2 , 3 , ... are called natural numbers. The set of all
natural numbers will always be denoted by the letter JV.
Natural numbers can be introduced axiomatically as follows. T h e
terms adopted as primitive a r e : Jf, 1, a n d the concept: m is the successor of n. The intuitive meaning of the last-named concept is that
m is that natural number which directly follows the number n. Thus, 2
is the successor of 1, 3 is the successor of 2 , etc.
The following axioms form an axiom system for natural numbers.
AXIOM I. 1 is a natural number, i.e., 1 e y T .
AXIOM IT. 1 is not the successor of any natural number.
AXIOM I I I . For every natural number η there is exactly
number m such that m is the successor of n.
one natural
AXIOM I V . If a natural number m is the successor of a natural number
n, and if m is also the successor of a natural number k, then η = k.
AXIOM V . T H E PRINCIPLE OF INDUCTION. If A is a subset of the
setJf
of natural numbers such that
(1)
(2)
le A,
for every natural number n: if n e A and m is the successor of
n, then m e A,
then every natural number is in A, so that A = Jf.
The axiom of mathematical induction is a mathematical formulation of the following intuition. Since I eA, hence, by ( 2 ) , the successor of 1 is in A, so that 2 e A. Hence, also in view of ( 2 ) , we infer
that the successor of the number 2 is in A. so that 3 e A. By reason-
28
II. NATURAL NUMBERS. PROOFS BY INDUCTION
ing in a similar way we conclude that 4 e A, etc. But we are not in
a position to repeat this reasoning infinitely many times. The principle of induction formulates the mathematical intuition which states
that if conditions (1) and (2) are satisfied, then every natural number
is in A.
All other concepts used in the arithmetic of natural numbers, such
as the operations of addition and multiplication, the relation "less
than", etc., can be defined by reference to the terms adopted above.
The axiom system given above, introduced by Peano in 1891, is sufficient to establish all those theorems of the arithmetic of natural numbers
with which we are concerned in mathematics *).
It will now be shown how the addition and the multiplication of
natural numbers can be introduced by reference to the primitive terms
adopted above. By Axiom III, for every natural number η there is
exactly one natural number m such that m is the successor of n. Let
the successor on η be denoted by ri. The addition of natural numbers
is defined thus :
(3)
n +1 =
(4)
n + m' = {n + m)'
ri
for every η
eJf,
for every η eJf
and m
eJf.
2
These two formulas form the inductive definition ) of the addition
of natural numbers. It will be demonstrated that, for every pair of natural
numbers n, m9 their sum n+m is defined in this way. Let η be any natural
number, and let A be the set of those natural numbers m for which
the sum n + m is defined. By (3), the sum n+l is defined, and hence
\eA.
Assume now that me A, i.e., that the sum n + m is defined.
By (4), the sum n + m' is defined, too, so that m' e A. We have thus
shown that assumptions (1) and (2) of the principle of mathematical
induction are satisfied for the set A. By this principle, every natural
number is in A, and hence the sum n+m is defined for every natural
*) Readers interested in proofs of arithmetical theorems based on Peano's axiom
should consult Grundlagen der Analysis by E. Landau, Leipzig 1930 (2nd ed. New
York 1946).
2
) Inductive definitions are widely used in mathematics on the strength of the
theorem on inductive definition (see Chapter XI, Section 6, and footnote *) on
p. 167).
1. AXIOMATIC APPROACH TO NATURAL NUMBERS. PRINCIPLE OF INDUCTION
29
number m. Since in the proof η stands for any natural number, the
sum n+m is defined for every pair n, m of natural numbers.
The following system of equations is an inductive definition of the
multiplication of natural numbers:
(5)
η· 1 = η
for every η EJV,
(6)
η · m' = (η · m)+n
for every η GJÎ and m GJV.
It will be shown that the product η · m is defined for every pair
of natural numbers n, m. Let η be any natural number, and let A be
the set of those natural numbers m for which the product n- mis defined.
By (5), the product η · 1 is defined, and hence 1 e A. If m G A, i.e.,
if the product n - mis defined, then, by (6), the product η · m' is defined,
too, so that /w' G A. We have thus found that the assumptions of the
principle of mathematical induction are satisfied by the set A. Hence
we conclude that every natural number is in A, so that the product
η · m is defined for every pair n, m of natural numbers.
The relation "less than" for natural numbers can be defined by
reference to addition as follows:
(7)
m < η if and only if there is a natural number k such that
m + k = n.
Other concepts used in the arithmetic of natural numbers can be
defined by reference to addition, multiplication, and the relation "less
than", and, as we have already pointed out, all the known theorems
in the arithmetic of natural numbers can be deduced from the axioms
adopted above.
Under the convention adopted in Chapter I, Section 5, the principle
of mathematical induction can be formulated thus:
1.1. If W is a property
and such that
defined on the set Jf of all natural numbers
(8)
(1 has the property
(9)
W(\)
W),
for every natural number n: if W(n), then W(n+\)
property W, then τζ+l has the property
W\
then every natural number has the property
W.
(if η has the
II. NATURAL NUMBERS. PROOFS BY INDUCTION
30
The principle of mathematical induction in form 1.1 is most often
used in proofs of mathematical theorems.
By making use of the principle of mathematical induction we shall
prove the following theorem called the minimum principle.
1.2. In every non-empty set of natural numbers there is a least number,
that is, a number which is less than, or equal to, any number in the set.
Assume that A is a non-empty set of natural numbers and that A
has no least number. Let Β be a set of natural numbers defined thus :
for every natural number η
η e Β ο for every natural number m, if m -ζ η, then m φ Α.
It can easily be seen that 1 G 2?, since otherwise the number 1 would
be the least number in A, contrary to the assumption.
Assume that n e B. It follows from the definition of the set Β that
for every natural number m, if m
then m φ A . Hence also n+1 φ A,
since otherwise n+l would be the least number in A, contrary to the
assumption. Consequently, « + l eB. We have thus demonstrated that
conditions (1) and (2) of the principle of mathematical induction are
satisfied by the set B. From this principle we infer that every natural
number is in B, so that, considering the definition of B, we conclude
that A is the empty set, which is contrary to assumption. The proof
of Theorem 1.2 is thus completed.
The following, logically stronger, form of the principle of mathematical induction follows from the minimum principle:
1.3. If A is a subset of the set Jf
of all natural numbers such that
(10)
le
(11)
for every natural number n: ifkeA
for all natural
k such that 1
then η + 1 G A,
A,
then every natural number is in A, so that A =
numbers
Jf.
Assume that A is a subset of Jf and that conditions (10) and (11)
are satisfied. Suppose now that there is a number η such that η φ A.
Thus the seLsf — A is a non-empty subset of Jf. By 1.2, the set Jf - A
has a least number n0. Obviously, n0 φ 1, since 1 G A. Every natural
number k which satisfies the condition 1 < k < n0 — \ is not in„V — A,
and hence is in A. It follows from condition (11) that n0e A, which
1. AXIOMATIC APPROACH TO NATURAL NUMBERS. PRINCIPLE OF INDUCTION
3 \
contradicts the assumption that n0eJf
— A. The assumption that
there is a natural number which is not in A has resulted in a contradiction. Hence we conclude that every natural number is in A, which
completes the proof of Theorem 1.3.
It is worth noting that the principle of mathematical induction in
the form of Axiom V is a logical consequence of Theorem 1.3. To
prove this assume that Theorem 1.3 is true and that A is any subset
of Jf which satisfies conditions (1) and (2) that occur in the formulation of Axiom V. Condition (1) coincides with condition (10). If condition (2) is satisfied, then condition (11) is satisfied a fortiori. Thus,
by applying 1.3, we conclude that every natural number is in A. This
proves that Theorem 1.3 implies Axiom V.
The following theorem often finds application in inductive proofs
of theorems :
1.4. If A is a subset of the set Jf of all natural numbers such that
(12)
leA
and
(13)
2eA,
for every natural number η > I: if n—XeA
n+1eA,
then every natural number is in A.
and ne A,
then
By (12), 1 eA and 2eA.
Let η be any natural number, η > 2 .
Assume that for every natural number k^n,
he A. In particular,
η— 1 e A and η e A. By (13), we infer that n+1 e A. Hence the assumptions of Theorem 1.3 are satisfied. By this theorem every natural number
is in A, which completes the proof of 1.4.
As shown above, the principle of mathematical induction can be
written in the form 1.1. Likewise, Theorems 1.3 and 1.4 can be reformulated by replacing the term "a subset A of the set Jf" by "a
property W defined in the s e t ^ T " and the symbols "m e A" by " W(m)".
Thus, for instance, Theorem 1.3 will assume the form
1.5. If W is a property
such that
(14)
(15)
defined in the set Jf of all natural
numbers
W(l)9
for every natural number n: if W(k) for all natural numbers k
such that 1 < k < n, then
W(n+l),
then every natural number has the property W.
Π. NATURAL NUMBERS. PROOFS BY INDUCTION
32
Theorem 1.4 will be reformulated analogously.
In view of the discussion carried out in Chapter I, Section 8, from
which it follows that every propositional function φ(χ), xeJf,
determines a subset A = [x eJf\ φ(χ)} of the set Jf of all natural numbers,
in the formulations of Axiom V and Theorems 1.3 and 1.4 reference
may be made t o the propositional function φ(χ), xeJf,
instead of
the subset A oîJf. Thus, for instance, Theorem 1.3, which is a logically stronger form of the principle of mathematical induction, yields
the following theorem:
1.6. If φ(χ) is a propositional function of the variable x, ranging over
the set Jf of all natural numbers, such that
(16)
(17)
φ(\) (1 satisfies the propositional function <p(x)),
for every natural number η : if (p(k) {k satisfies φ(χ)) for all natural
numbers k such that Λ ^k^n,
then φ(η+1) ( « + l satisfies
<p(x)),
then every natural number satisfies the propositional function
φ(χ).
Axiom V and Theorem 1.4 may be similarly modified.
2. Examples of proofs by induction
Those proofs of mathematical theorems in which use is made of
the principle of mathematical induction are called inductive. Examples
of such proofs applied t o combinatorial problems will be given below.
2.1. For every natural number n, the number of all subsets of a set
n
of η elements equals 2 .
Let W be a property of natural numbers such t h a t : W(n) if and
n
only if the number of all subsets of a set of η elements equals 2 . We
find that
(1)
the number 1 has the property W.
For let Al = {a^. This set has two subsets: the empty set Ο and
the set At = { a j . Hence the number of all subsets of a set of one
1
element equals 2 = 2 .
2. EXAMPLES OF PROOFS BY INDUCTION
33
It will be proved now that
(2)
if a natural number η has the property W, then n +1 has the
property W.
T o do so we assume that the number η has the property W. Con= {al9
an, an+1}.
We
sider now any set of n + 1 elements, An+l
divide all the subsets of that set into two families. The first family will
contain every subset which does not contain the element an+l,
whereas
the second will contain every subset in which an+l does occur. These
families are disjoint, and their union is the family of all subsets of the
set An+1.
The first family contains all the subsets of the set
An = {ax, ..., an) of η elements, and does not contain any other set.
Since by assumption η has the property W9 the first family contains
exactly 2" sets. The sets in the second family are obtained from sets
of the first family when the element an+l is joined to a set of the first
family. The second family does not contain any other sets, hence it
has as many sets as the first family does, i.e., 2". Hence the number
n
n
n+1
of all subsets of Λ Π + 1 equals 2" + 2 = 2 · 2 = 2 .
Hence we conclude that the number η + 1 has the property W9 which proves condition (2).
As the assumptions formulated in 1.1 are satisfied, we infer from
1.1 that every natural number has the property W, which was to be
proved.
The following notation will now be adopted: for any pair of natural
numbers n, k, such that k < η, we set
n(n-l)
(3)
...
(n-k+l)
F i - ... · k
The symbols
are called Newtoris symbols. N o t e that in the numer-
ator of the right-hand side of equation (3) there is the product of k
successive decreasing natural numbers, beginning with n, and in the
denominator, the product of k successive increasing natural numbers,
beginning with 1.
It can easily be shown that the following equation holds:
(4)
Π. NATURAL NUMBERS. PROOFS BY INDUCTION
34
This is true because, by Definition (3), we have
_ « ( w - l ) ... (/ι-Α:+1)
~
_
1-2
, ? ( w - l ) ... (n-k
:
h
- ... /c
... (n-k
~
1 -2-
+ 2)(n-k
1 -2-...-(A-
+
+ 2)k
-1)1
\+k)
...
_ (n+\)n(n-\)
... (n-k
""
1 ·Τ· ...·£
+ 2)
/«H h
\ Α: Γ
:
Let ^ n =
. . . , 0 „ } be any set of « elements. Every subset of
this set, consisting of k elements, will be called a combination of k elements, where 1 · k < η.
The following theorem determines the number of all combinations
of k elements from the elements of a set of η elements (k < n).
2.2. For every natural number η and k < n, the number of all combinations of k elements from the elements of a set of η elements equals j ^ j .
Now let W stand for a property of natural numbers such that:
W(n) if and only if the number of all combinations of k elements,
k < «, from the elements of a set of η elements equals Q . It will be
shown that
(5)
1 has the property W.
If η = 1, then k can only equal 1. The number of all combinations
of 1 element from the elements of a one-element set equals 1. Thus
^1
= 1. Hence 1 has the property W.
It will now be proved that
(6)
if η has the property W, then η + 1 has the property W.
To prove this assume that a natural number η has the property W.
Consider now any set of n+l elements, An+l
= {al, ..., an+1},
and
form all the combinations of 1 element. Their number is obviously
EXERCISES
n+ l =
35
Note also that the number of all combinations of
equals 1
elements from the elements of
n+l
Now let k be
any natural number such that 1 < k < n+ 1. Now form all the comi.e., all the subsets
binations of k elements from the elements of An+l9
of k elements each, and distribute them into two families of sets. Let
the first family contain every subset of An+1 of k elements in which
an+1 occurs. The second family will contain all those subsets of
An+i
in which an+i does not occur. These families are disjoint, and their
union is the family of all subsets of An+1 of k elements. The first family
will have as many sets as subsets of k— 1 elements that can be formed
from the set {ax, . . . , # „ } of η elements. Since η has the property W
by assumption,
J such subsets can be formed. The second family
has as many sets as subsets of k elements that can be formed from the
set {al9
an} of η elements, i.e.,
as η has the property W. The
number of all combinations of k elements formed from the elements
of An+l
is the sum of the number of sets in the first family and the
number of sets in the second family, which by (4) comes to
(.:,κκ:')·
We have thus shown that for every 1 < k < n+\ the number of
all combinations of k elements formed from the elements of An+X equals
and hence η +1 has the property W. This completes the proof
of condition (2). Thus the assumptions formulated in Theorem 1.1
are satisfied. On the strength of that theorem we conclude that every
natural number has the property W. Hence Theorem 2.2 is proved.
Exercises
1. Prove that if A is a subset of the set Jfoî
(1)
2eA,
all natural numbers such that
36
II. NATURAL NUMBERS. PROOFS BY INDUCTION
(2)
for every natural number n: if η e A, then η+2 e A,
then every even natural number is in A.
2. Prove that if A is a subset of the set jVof all natural numbers such that, for
a fixed natural number n0,
(1)
n0eA,
(2)
for every natural number η : if η > n0 and η e A, then n+1 e A,
then every natural number η > nQ is in A.
3. Prove by induction the following formulas :
( Λ - 1 ) · Λ · (/Z+1)
1 - 2 + 2 - 3 + ... + ( « - » • / ! =
3
2
3
1,
îorn>
3
( 1 + 2 + ... +n) = l + 2 + ... + * .
4. Prove by induction the following inequalities :
n
rt\> 2
n
for 7f>4;
<
2
for Λ > 4;
2 > rt
(=±i)-.
5. For any natural numbers m, η the symbol m\n stands for: m is a divisor of //.
Prove by induction that, for every natural number /?,
3
5
3|n -/î,
5|rt -rf,
Ί
Ί\η -η.
k
6. Determine whether the theorem: k\n — n is true for all natural odd numbers
k and for every natural number n.
7. Prove by induction Bernoulli's inequality (1689):
n
(l+a)
> 1+rttf,
a> - 1 .
8. Prove by induction Newton's binomial theorem :
(a+b)
n
= ö"+ Qa"-^+ Q a " - 6 +
2
2
... + Q ß - * 6 + ... + ^ " j ab ~ + b .
n
fc
n l
n
j
Hint. In the proof make use of formula (4) in Section 2.
9. Let ûi = 1, a2 = 1, tfn+2 = α π + ι + β Λ {Fibonacci's sequence). Prove by induction
the following theorems :
2 | α 3 π,
3 | Λ 4Π ,
5|α 5,,.
10. Prove by induction that Fibonacci's sequence, as defined under 9, above, can
be defined as follows :
CHAPTER III
FUNCTIONS
1. The concept of function*)
Let X and Y be any non-empty sets. If for each element of X there
is exactly one element of Y associated with it, then we say that a mapping of the set X into the set Y has been established. Instead of speaking
of a mapping of X into Y we also say about a transformation of X into
Y or about a function which maps (transforms) X into Y. Thus the terms:
mapping, transformation, and function have one and the same meaning.
Mappings of one set into another will usually be denoted by the
letters / , g, h. The formulation " / is a mapping of a set X into a set Y"
will be replaced by the symbolic formula
(1)
f:X-+Y.
If / : X -> Υ, then that element of Y which has been associated by
the mapping / with an element Λ: of A" will be called the value of the
function f for the argument χ or the image of the element χ under the
mapping f and will be denoted by f(x). Under this convention the
symbols / and f(x) have different meanings. The former stands for the
function itself, that is, for the association with the elements of X of
elements of Y. The latter denotes the value of the function for the
argument x, that is, an element of Y.
The symbol fx will sometimes be used for f(x). The function whose
values for the arguments x e l a r e elements fxeY
will sometimes be
denoted by
(fx)xeX.
When defining a function we employ the following sets: a set X of
*) In Chapter III the concept of function is treated intuitively. A rigorous definition of a function, formulated in terms of concepts of set theory, will be given
in Chapter V, Section 5.
HL FUNCTIONS
38
those elements for which the function has been defined, called the set
of arguments of that function or the domain of that function; a set Y,
which contains the values of the function; and a set of the values of
the function, i.e., the set of those elements y of Y for which there is
an x e X such that y = f(x). The set of values of a function / is
called the codomain of / and will be denoted by f(X). It follows from
the definition of the set of values of a function that, for every y e Υ,
(2)
(y ef(X))
ο (there is a n j c e l such that y =
Instead of there is an χ eX such that the symbol V
used. Then (2) can be written in symbols t h u s :
(3)
f(x)).
will often be
xeX
(yef(X))o\J(y=f(x)).
Examples
The function / : M -> 0t (where 0t is the set of all real numbers),
defined by the formula f(x) = sin χ is a function which maps the set of
real numbers into the set of real numbers. Thus its domain is the set
0t. Its codomain i s { j > e ^ : — 1 < j> < 1}, that is, under the convention adopted in Chapter I, Section 8, (5), the set of those real numbers
y which satisfy the propositional function - 1 < y < 1. This is so because we conclude from (3) that a real number y is in f($) if and only
if there is a real number χ such that y = s i n * . This condition is satisfied if and only if — 1 < y *ζ. 1. As a second example consider the func2
tion f\0t-±0t
defined by the formula f(x) = x . Its domain is the set
0t of real numbers, and its codomain, the set {y e 0ί\ 0 < y}, that is,
the set of non-negative real numbers. This is so because it follows from
(3) that a real number y is in / ( 0ί) if and only if there is a real number
2
χ such that y = χ . This condition is satisfied for the non-negative real
numbers, and for those numbers only.
Concerning the f u n c t i o n s / : X - • F i t is also said that they are defined
on X with values in Y.
x
The set of all the functions f: X -+ Y will always be denoted by Y .
We say that a function f.X-*Y
maps X onto Y if the codomain f(X)
o f / c o i n c i d e s with Y, i.e., iff(X) = Y. Since f(X) cz Y, hence the equation f(X) = Y holds if Y czf(X), i.e., if every element y of Y is an
1. CONCEPT OF FUNCTION
39
element off(X). This condition is satisfied if and only if for every y eY
there is an x e X such that y = f(x) (every element of Γ is a value of / ) .
For instance, the function f: 01
0t defined by the formula f(x) = sin χ
maps 0t into 0t, but it maps 0t onto {x e 0t\ - 1 < J C < 1} = f(0t).
2
Likewise, the function / : 0t -» 0i defined by the formula f(x) = x
maps 01 into 0t, but it maps 0t onto {x e 0t: 0 ^ x} = f(@).
A function / : X
Y is equal to a function g : X
Y if, for every
χ eX, the condition f(x) = g(x) is satisfied. Instead of for every χ eX
we shall often write / \ . Then the condition which is necessary and
xeX
sufficient for two functions / : X -> Y and g: X -> Y t o be equal can be
written in symbols t h u s :
(4)
(/=*)<>Λ(Λ*)
=*(*))·
xeX
F o r instance, the functions f\ffi-+0t
and g\ 0t -+ 0t defined, re2
spectively, by the formulas f(x) = x — 1, g(x) = (x— 1)(JC+ 1) are equal.
N o w let / : Ζ
F be a mapping of A" into Y. It is sometimes necessary t o examine the function / not on the entire set X, but on a subset
A cz X. We then define a new function g: A -> Γ as follows:
(5)
g(x) = / ( * )
for χ Ε A.
This new function g is denoted by f\A and is called the function f
restricted to the set A. Its domain is obviously A. For instance, by restricting the function f\0t-+0t
defined by the formula f(x) = c o s * for
those χ which satisfy the condition — π < χ < π we obtain the function
g =f\{x e 0ί\ — π ^ χ < π } , that is, with the cosine function restricted
to the set {x e 0ί\ —π < χ < π } .
%
Let g : v4
Y be defined on a subset A of A", and l e t / : A' - • 7 . If
/ | Λ = g , then it is said that / is an extension of g to X. For instance,
let g : Q
{0, 1} (where Q is the set of all rational numbers) be a function defined by the formula g(x) = 1 for x e Q , and l e t / : 0t
{0, 1}
be a function defined as follows:
JO
f
(
X
)
=
=
\l
f o r x e ^ - Q ,
forxeQ.
The function / (called the Lejeune-Dirichlet
of the function g.
function) is an extension
III. FUNCTIONS
40
Certain kinds of functions / which m a p a set X into a set Y will be
discussed. If Y is the set of real numbers, then the function / : X -+ Y is
called a real function. If Y is the set of complex numbers, then the function / : X -* Y is called a complex function. If Z is the set of natural
numbers, then the function / : X -> Y is called an infinite sequence or
a sequence. The value /(w) of a function / for an argument « is called
the n-th term of a sequence. The n-th term of a sequence is usually denoted by an9 and the sequence itself, i.e., the function, by (an) or by
(al9a2,
..·)· If X = { 1 , 2 , ...9k}9
then the function / : X
F i s called
a y&iz'te sequence of k elements. A finite sequence is usually denoted by
(al9
. . . , f l f c) -
The following theorem will now be proved:
1.1. If X is a set of η elements, and if Y is a set of m elements, then
n
x
n
there are m functions which map X into Y, so that Y has m elements.
Assume that m is any natural number, and that Y = {bl9
bm}.
The proof will be by induction with respect t o n. Let η = 1 and let
1
X = { f l j . There are then exactly m = m functions which m a p X into
F a n d which are defined as follows: f(x) = bx, ... 9fm(x) = bm for χ eX.
n
It will now be shown that if there are exactly m functions which m a p
n +1
the set {ai9
of η elements into Y9 then there are exactly m
functions which map X = {a1, ..., an9 an+l} into Y. We now split up the
set of all mappings of X into Y into m disjoint sets: the first set will
contain all those mappings / : X - • y for which f(an+1)
= bx, the second,
all those mappings / : X -+ Y for which f(an+l)
= b2, the m-th set, all
those mappings / : X ^ Y for which
= bm. The set of all the
mappings of X into Y is the union of these m sets. Each of them contains as many mappings as can be formed by mapping {al9 ..., an} into
n
Y9 i.e., by assumption, m . Hence we conclude that the number of all
n
n+1
mappings of X into Y is m · m = m . Under the principle of induction we conclude further that for every natural number η there are
n
exactly m mappings of a set of η elements into a set of m elements.
Since m is assumed to be any natural number, the proof of Theorem
1.1 is completed.
It follows immediately from 1.1 that
n
1.2. If Y is a set of m elements, then there are m sequences of η
elements each whose terms are in Y.
2. ONE-TO-ONE FUNCTIONS. INVERSE FUNCTION
41
2. One-to-one functions. Inverse function
A function / : X -> Y is called a one-to-one function if it takes on
different values for different arguments. This condition can be written
in symbols thus :
Λ
(1)
Λ Ol
# * 2 =>f(Xl)
#/K*2)).
Examples
The function f: 0i ^> 0t (where ^£ is the set of real numbers) defined
by the formula f(x) = \x\ is not a one-to-one function, because / ( l )
= / ( — 1) = 1. The function g: Jf - » Q (where Q is the set of rational
numbers) defined by the formula g(n) = l/n is a one-to-one function,
e
for if « 7 m, then 1/« # \\m for any pair «, m of natural numbers.
A function g: Y -* X is called inverse to a function / :
-> Γ if Γ
= / ( X ) (the set of arguments of g is the set of values of / ) , X = g(Y)
(the set of arguments o f / i s the set of values of g), and for every χ eX
the following equation holds:
(2)
g (Ax))
= *·
It will be proved that
2.1. If a function g: Y -* X is inverse to f:X
the condition
(3)
f(g(y))
Y, then for every y e Y
= y
is satisfied.
Let y be any element of Y. As g is inverse to / , it follows from the
definition of the inverse function that Y = f(X). Hence y e f(X). We conclude from formula (3) in Section 1 that there is an element χ eX such
that y = f(x). This and (2) yield the equations
f(g(y))=f{g(f(x)))=f(x)
= y-
2.2. If g: Y -+ X is inverse to f: X -> Y, then the following
are satisfied:
(4)
(5)
(6)
conditions
/ maps X onto Y and g maps Y onto X,
(g(y) = x)o
(f(x)
= y)
for every x e X and y
/ and g are one-to-one
eY,
functions.
Condition (4) follows from the definition of the inverse function.
III. FUNCTIONS
42
We have f(X) = Y, which means that / maps X onto Y, and at the same
time g(Y) = X, which means that g maps Y onto X.
To prove (5) assume that g(y) = x. This and (3) yield that
f(x)=f(g(y))=y.
Hence if g(y) = x, t h e n / ( χ ) = y. Conversely, if f(x) = y, then under
(2) g(y) = g(/W) = *> so that g(y) = x.
T o prove (6) assume t h a t / ^ ) = f(x2). Then gÇ/ixJ) =
g(f(x2)\
From this and from (2) we conclude that xx = x2. Hence if *i ^ x 2,
then /Οι)
f(xi), which proves t h a t / i s a one-to-one function. Assume
= f(g(y2)).
This and (3) yield
now that g{yx) = g(y2). Then f{g{yù)
yx = y2. Hence if yx φ y2, then gOO Φ g{y2\ which proves that g is
a one-to-one function.
It will now be proved that
2.3. For every one-to-one function f. X -+ Y which maps X onto Y
there is exactly one inverse function.
Assume t h a t / : X
Y satisfies the conditions specified in the theorem.
Since / maps X onto Y, hence for every y e Y there is an ^ e l such
that y = f(x). At the same time, as / is a one-to-one function, for
a given y e Y there can be at most one χ such that y = f(x). For should
a n
e n
we
y = f(*i)
d y = /C*2)j t h
would have f(xi) = f(x2), and this,
together with (1), would yield JC X = x2. Consequently, for every y e Y
there is exactly one xeX
such that y = f(x). Let g: Y -> X be the
function defined thus:
(7)
for every y e Y, g(y) = χ if and only if f(x) = y.
The function g is the inverse function of / . Of course, g maps Y into
X. S i n c e / m a p s X onto 7, hence f(X) = Y. At the same time g(Y) = X,
since for every x e X there is an y e Y such that y = f(x), and this is
equivalent to the condition λ* = g(y). Condition (2) is satisfied, too. For
let f(x) = y. Then, by (7), g(/(x)) = g(y) = x. We have proved that for
every one-to-one function f: X -> Y which maps X onto Y there is an
inverse function. Theorem 2.2 (5) and Definition (7) of the inverse function g imply that for a given one-to-one function which maps X onto
Y there is only one inverse function: This completes the proof of Theorem 2.3.
- 1
The function inverse t o / w i l l be denoted b y / .
2. ONE-TO-ONE FUNCTIONS. INVERSE FUNCTION
43
Examples
Let / : 0t -> 0t be a function defined by the formula f(x) = 2x. This
is a one-to-one function which maps 0t onto 0t. The inverse function
is defined by the formula
= y*- Let g: ^ + -» ^ + be a function
which maps the set 0t+ of non-negative real numbers into 01+ and is
2
defined by the formula g(x) = x . N o w g is a one-to-one function and
-1
it maps 0t+ onto 0t+. The inverse function g is defined by the formula
_1
+
g (jc) = ]/x. It also maps @ onto 0t+.
The following equations follow from Theorem 2.3 and from (2) and
(3):
(8)
f-\Ax))
= x
forxeX,
= y
former,
where X is the domain of/;
1
(9)
f(r (y))
where Y is the codomain of / .
N o t e also the next result that follows easily from the definition of
the inverse function and from equation (2).
2.4. If g is the function inverse to f
then f is the function inverse to g.
Hence the following equation holds :
(10)
=/.
One-to-one mappings of a set X onto X will be called permutations
of X. The simplest example of a permutation of X is the identity function
Ix: X -+ X, defined by the formula
(11)
/*(*) = *
forxeX.
Let η ! (to be read : the factorial of n) denote the product of the successive natural numbers from 1 to η :
(12)
ni = 1 · 2 · ... · λ .
It will be proved that
2.5. The number of all one-to-one mappings of a set of η elements onto
a set of η elements equals n\. Accordingly, the number of all permutations
of a set of η elements equals n\.
III. FUNCTIONS
44
If Χ = {ax} and Y = { è j , then the number of all one-to-one mappings of X onto Y equals 1 = 1!. This is so because the only such mapping is / : X -» Y, where /(a^ = bx. We show that if the number of all
one-to-one mappings of a set of η elements onto a set of η elements
equals n\, then the number of all one-to-one mappings of a set of n + 1
elements onto a set of « + l elements equals (« + l)!. Consider the sets
X = {al9
tfn+1}
and Y = {bl9 ..., ό„, bn+l}.
The set of all one-toone mappings of X onto Y will be divided into η + 1 disjoint sets as
follows: the fc-th set (k = 1, . . . , w + l ) will contain every one-to-one
mapping of X onto Y, f: X
Y9 for which f(an+1)
= bk. For every
k = 1, ..., n +1, in the /c-th set the number of all one-to-one mappings
...,#„}
equals the number of all one-to-one mappings of the set {al9
onto the set {bi9
bn+i} — {bk}, that is, n\ by assumption. This is so
because for every function / in the k-ih set the value f(an+1)
= bk is
fixed, and so only the values of the function for those arguments which
are in {al9
an} may be fixed arbitrarily, and those values will have
to be in the set {bl9 ..., bn+i} — {bk}; only one-to-one functions are considered. Since in every set there are n\ functions, and the number of
sets is « + l , hence the number of all one-to-one functions which m a p
X onto Y is n\- (n+l) = 1 - 2 - ... ·/i · (/i + l) = ( / i + l ) ! . By applying
the principle of mathematical induction we complete the proof fo
Theorem 2.5.
Example
Let X = { 1 , 2 , 3}. By Theorem 2.5, the number of all permutations
of X equals 3 ! = 6. We now write out all the permutations of that set:
fi9i=
1,2,3,4,5,6;
/ i d ) = 1,
/i(2) = 2 ,
Λ (3) = 3 ,
Λ(ΐ) = 1,
Λ(2) = 3 ,
Λ(3) = 2 ,
Mi)
= 2,
Λ(2) = 1,
Λ(3) = 3 ,
AO)
= 2,
Λ(2) = 3 ,
Λ(3) = 1,
MD
= 3,
Λ (2) = 1,
/ 5( 3 ) = 2,
Mi)
= 3,
Λ(2) = 2 ,
ΛΟ) = 1.
3. COMPOSITION OF FUNCTIONS
45
3. Composition of functions
Given the functions / : X -> Y and g: Υ -» Ζ , for every χ eX there
is exactly one z e Ζ such that ζ = g(f(x)). Thus the functions / and g
determine a new function A: X
Z, defined t h u s :
A(x) = g ( / ( x ) )
(1)
for every χ
eX.
The function A is the composition of the functions f and g and is denoted by g ο / . Thus by definition we have
(g o / ) (x) = g ( / ( x ) )
(2)
for every
xeX.
Examples
Let / : ^ -> ^ be a function defined by the formula / ( x ) = 2x, and
let g: ^ -> 0t be a function defined by the formula g(x) = cos*. Then
(g° f) (x) — cos2x for x G 01. Let now / be a function that maps the
set of complex numbers into the set of non-negative real numbers and
is defined by the formula f(z) = |z|, and let g be a function that maps
the set of non-negative real numbers into the family of all subsets of
the set of real numbers and is defined by the formula g(y) = { x e l :
—y < x < y}. Then (g of) (z) = [x e St: -|z|
. χ <: |z|} for every
complex number z.
The following theorem states that the composition of functions is an
associative operation.
3.1. For any functions f: X -> Y, g : Υ -> Ζ , and h: Ζ -* W
(3)
h o (g of)
= (A o g)
of.
It follows from the definition of the composition of functions that,
for every χ e X9
(A ° ig » / ) ) (x) = h((g of) (χ))
=
h(g(f(x))),
((A ° g) °/) W = (h°g)
=
h\g(f(x))).
(fix))
Formula (3) follows from the two equations above.
Note that the composition fog need not exist for all those functions
f: X -* Y and g: Υ -» Ζ for which the composition g o / does exist. The
composition fo g cannot be defined if the set of values of the function
g is not contained in X. For the f u n c t i o n s / : X -> Y and g: Y
Ζ there
are compositions g o f\ X
X a n d / o g: y
Κ In particular, if X = 7,
III. FUNCTIONS
46
then gof: Χ -> X and f ο g: Χ -> Χ. Usually, however, g°f φ f o g. For
2
2
instance, if f(x) = sinx and g(x) = x , then (g of) (χ) = ( s i n * )
2
2
= s i n * , and (/© g) (χ) = sin χ . These two functions are different.
It follows from the definition of the composition of functions and
from equations (8), (9), (11) in Section 2 that
3.2. For every one-to-one function f that maps X onto Y
1
(4)
f~ of=Ix
and
fof-*=IY9
where Ix is the identity function defined on X, and IY is the identity function
defined on Y.
We now prove a theorem on the composition of functions that m a p
certain sets onto certain sets and on the composition of one-to-one functions.
3.3. For any functions f: X -> Y and g: Y -> Z,
(5)
/ / / maps X onto Y and g maps Y onto Z, then g of maps X onto Z,
(6)
/ / / and g are one-to-one functions,
tion,
0)
If f d g are one-to-one functions which map, respectively, X and
x
l
1
Y onto Y and Z, then the functions (g ο f)~ andf~ o g' exist, and
1
1
1
(gof)=f~ ogholds.
then go f is a one-to-one func-
an
T o prove (5) assume that / maps X onto Y and g maps Y onto Z. Let
ζ be any element of Z. There is then y G Y such that ζ = g(y). At the
same time there is an χ eX such that y = f(x). Hence ζ = g(f(x))
= (g ο / ) (χ). Thus gof maps X onto Z.
T o prove (6) assume that / and g are one-to-one functions. If the
condition xx φ x2 is satisfied for x^eX and x2 e X, then f(Xi) Φ f(x2)Consequently, glfix^) Φ g(f(x2)),
so that (g of) (*,) φ (g of) ( χ 2 \
which proves that g o fis a one-to-one function.
It follows from ( 5 ) and (6) that if / and g are one-to-one functions,
if / maps X onto Γ, and if g maps Y onto Z, then g ο f is a one-to-one
function that maps X onto Z. It follows from 2.3 that there is a function
inverse togof
which will be denoted by (gοf)~\
and which maps Ζ
1
onto X. At the same time there are functions: g" : Ζ -> Y, which maps
1
Ζ onto Y, and f' : Y -+ X, which maps Y onto X. Thus there is the
4. GROUPS OF TRANSFORMATIONS
47
- 1
-1
_ 1
c o m p o s i t i o n / o g : Ζ -* X, which maps Ζ onto X. Let ( g o / ) ( z )
= x; then ( g o / ) (χ) = ζ in view of 2.2 (5), so that g(f(x)) = z. Let
_ 1
_1
/(*) = y; then g(>>) = z. Consequently, / ( y ) = χ and g ( z ) = >\
1
- 1
- 1
- 1
l
-1
Hence ( Γ o g ) (z) = / ( g ( z ) ) = f~ (y)
= χ. Conversely, if ( / ο
1
- 1
- 1
-1
1
o g " ) (z) = x, t h e n / ( g ( z ) ) = χ Let g ( z ) = >>; t h e n / " ^ ) = x.
Consequently, f(x) = y and g(y) = z. Hence (g ο / ) (χ) = g(/(x)) = g(y)
H
= z. In view of the above ( g o / ) ( z ) = x, which completes the proof
of (7).
It follows in particular from Theorem 3.3 that
3.4. A composition of two permutations of X is also a permutation of X.
4. Groups of transformations
Let Χ Φ Ο be any non-empty set. Let ^ be the set of all permutations of X, i.e., the set of all one-to-one mappings of X onto X. Let ©
be any subset of
© is said to be a group of transformations
if the
following conditions are satisfied:
(gi)
the identity transformation of X is in ©,
(g 2 )
if / G © and g e ©, then the composition g ο / e ©,
(g 3 )
if / e ©, then the inverse transformation f~
l
e ©,
which in the symbolic notation is written as follows:
Ixe
(gi)
(gl)
to)
(fe
©,
© A g e ©)=^(go/e
fe
© =>/
- 1
©),
e ©.
Examples
The set of all permutations of any set Χ Φ Ο is the simplest example
of a group of transformations. For instance, the set of all permutations
of the set { 1 , 2 , ..., n) forms what is called a symmetric group Θ„. The
set of all permutations of the s e t J ^ of all natural numbers forms what
is called an infinite symmetric group. Let now © be a set which consists
of the following two one-to-one transformations of the set 0t of real
numbers onto 0t \ the identity transformation I& and the transformation
/: 0t -+ 0t defined by the formula f(x) = —x. The set © is a group
ΠΙ. FUNCTIONS
48
of transformations. Condition (g x ) is self-evidently satisfied. Condition
(g 2 ) is also satisfied, since, as can easily be verified, the following equations hold:
Ia°Ia
= Ia,
°/= / ° = / >
f°f=I&,
which prove that the composition of two transformations which are in
1
1
(9 is also in ©. Condition (g 3 ) is satisfied, because I^ =
and f' = f,
as may easily be verified. Let © be a set of transformations
(
{ €—the set of all complex numbers) defined by the following formula:
(1)
Ax,y)
= (x + a,y + b),
where a and b are any real numbers. It can easily be verified that those
transformations which satisfy condition (1) are permutations of
and
that conditions (gj), (g 2 ), (g 3 ) are satisfied. Thus © is a group of transformations.
Exercises
1. Let / b e a mapping of the set M of all real numbers into the family of all subsets
of 01, defined by the following formula:
/(/)=
F i n d / ( - l ) , /(Ο),
|/|}
{xe0t\x<
for every te
at.
2
f(t +\).
2. Determine the domain (a subset of the set M of all real numbers) of the function / defined by the formula
f(x) = ]/logsinx.
3. Let / : 0t-> M be a function defined by the formula f(x) = — 2 * 4 - 3 , and let
l
l
A = { — 4, — 2 , 0 , 2, 4 } . Determine the inverse function f~ and find f~ (A).
4. Let / be the function that maps the set X = {xe & :0 < χ < 2} into 01 defined by
\\ίχ
for 1 < χ < 2.
Determine f(X) and define on f(X) the function inverse to / .
5. Give an example of a mapping of the set {x e 0t : χ > 0} onto the set
0 < χ < 1}.
{xe
6. Let us denote by
η
the permutation of a set A = {1, 2,
number ak (k = 1, ..., ή).
2
...
»j
a2
...
anl
n} which associates with the number k the
EXERCISES
Let/= ^
,
=
ι 4 3/ ^ \i
2 4
) ·
3
F i n
d
49
t he
c o m
P°
s i t i o n 0s
/ ^
a
nd
7. How many functions, such that / ( l ) = 3, map the set {1, 2, 3} onto itself?
Generalize the result for the case in which the set in question is {1, 2, ..., n} and the
functions/are such that/(*i) = yi9 i = 1, 2, ..., k\ k < n\ xiyyi e {1, 2, ..., ή).
8. Prove that if X is a set of k elements and if F is a set of η elements, k < n,
then there are n(rt— 1) ... (n—k+l) one-to-one functions that map X'mio Y.
9. Prove that if / is a mapping of a set A onto A such that / © / = / , then / is an
identity mapping of A onto A.
1 0 . Let © be a group of transformations / a of the set # of all complex numbers
into
defined by
f<x.(x+iy) = *cosa—>>sina—/(xsina-f-ycosa),
where a is any angle. Prove that © is a group of transformations.
CHAPTER IV
GENERALIZED U N I O N S A N D INTERSECTIONS OF SETS
1. The concept of generalized union and intersection
Let Χ Φ Ο be any space and let 9\ be the family of all subsets of X.
Let also Τ φ Ο be any set. A function / : Τ
91 will be called an indexed
family of sets (more exactly: an indexed family of subsets of X). Let
f(t) = At for t e T. Obviously, At e9l, and hence At is a subset of X
for every / e T. Under the convention adopted earlier (cf. p. 37) the
function / will be denoted by
(At)teT.
Examples
(I) Let Τ = {1, 2 , 3 , 4 , 5} and let 91 be the family of all subsets
of the s e t e / f of natural numbers. Assume
f(t)
Then (At)teT
subsets of^T :
= At = {m eJf\
t <m)
for t e T.
is the indexed family consisting of the
A, = { 2 , 3 , . . . } ,
A2 = { 3 , 4 , . . . } ,
AA = { 5 , 6 , . . . } ,
following
A3 = { 4 , 5 , . . . } ,
A5 = { 6 , 7 , . . . } .
(II) Let Τ = Jf and let 91 be the family of all subsets of the set 01
of real numbers. Assume
f(n) = An = j j c e 3t\ ( - 1 ) " < χ < 1
for ne
jr.
In this case the indexed family (An)nejr
is a sequence of subsets
of the set of real numbers. The first four terms of that sequence are
written out below:
Ai = {x e ®\
-1
Α2 = {xe
1 < χ < 1+y},
ίζ
χ ί ζ 2},
1. CONCEPT OF GENERALIZED UNION AND INTERSECTION
A3 = {xe
^
0t\
= {xel:
(Ill) Let Τ = 0t (0t—the
-1
51
< x <
1< χ <
set of real numbers) and let 91 be the
family of all subsets of 0t. Let
m\ \x\
f(t) = At = {xe
2
</ }.
is an indexed family of certain subsets of 0t.
The indexed family (At)teT
Some of the sets At, teT,
will now be written out:
A-il
= {x e 0l\ \x\ < 2 } ,
Λ_3 = { x e l :
Let (At)teT
|JC| < 9 } ,
= {x e 0t\ \x\ < 1/4},
A1/2
Λ0 = {xe ^ :
|JC| < 0 }
= {0}.
be any indexed family of subsets of a space X.
set, denoted by [J At,
The
t e T, which contains an element χ of X if and
teT
only if Λ: is in at least one of the sets At,
of sets At, t e T. In other words, x e\J
such that Λ: e At.
and only if there is a, t e Τ
This can be written in symbols as follows:
(xe{J
At)o\J(xeAt).
teT
{1,2,
(2)
At\ï
teT
(1)
If Γ =
for t e T, is called the union
teT
then
LMr
= ΛιυΛ2υ
... υ Λ „ *).
/er
Ιΐ Τ =JV
(where Jf
is the set of natural numbers), then the fol-
lowing notation is used:
(3)
\JAn
instead of
η= 1
*) If Τ = {1, 2,
\JAU.
nejr
w}, then the indexed family of sets {At)teT
The generalized union U
teT
is then the union AiVA2v
= (Alt A 2 , ..·, An).
... \JAn in the sense of
the definition adopted in Chapter I, Section 2. More generally, if Γ is a finite set,
then the family (At)teT
is a function whose set of values consists of a finite number
of sets. In this case the generalized union U A t is the union (in the sense formulated
teT
by the definition in Chapter I, Section 2) of the sets which are values of the function (At)teTThe concept of generalized union of sets, as introduced in this Section, thus generalizes the concept of union, as defined in Chapter I, Section 2,
and includes the case in which the set Τ is infinite.
IV. GENERALIZED UNIONS AND INTERSECTIONS OF SETS
52
Examples
Let (At)teT
\jAt
be the indexed family of sets defined under (I). Then
= A1UA2KJA3UA^UA5
= { 2 , 3 , ...}
=^-{1}.
teT
For the indexed family
we have:
(4)
(sequence) of sets defined in (II)
(An)nejr
- 1 < * < 2 } .
\J An = {xeM:
N= 1
This equality holds because: 1° if x e {x e 0t\ — 1 < χ < 2 } , then
χ e Al9 and hence χ is in at least one of the sets An for η eJf\ 2° if χ is
n
in at least one of the sets An9 then the condition (— l ) < χ < 1 + 1/«
is satisfied for this n9 and hence it follows that x e {x e £%: — I < x < 2 } .
F o r the indexed family (At)te<% as defined in (III) we have
\jAt
=
0t.
tem
2
For if χ e ^2, then there is a positive real number / such that \x\ < t .
Hence χ e At. On the other hand, if χ e At for some t e &9 then χ is
a real number, and thus χ e 01.
Let (At)teT
be any indexed family of subsets of a space X. It follows
from the definition of the union of sets At9
teT,
that an element χ
of X is not in U At if and only if the condition that Λ: is in at least
teT
one of the sets At is not satisfied, that is, if and only if χ is not in any
of the sets At9 teT.
This is written in symbols t h u s :
(5)
At)o
(χφυ
teT
Let (At)teT
A(*tAt).
teT
be any indexed family of subsets of X. The set, denoted
by ΓΊ At9 which contains an element χ of X if and only if χ is in each
teT
of the sets A T 9 for t e T9 is called the intersection of sets A T 9 teT.
other words, x e D A T if and only if χ e A T for every teT.
written in symbols as follows:
(6)
(xer\A )o/\(xeA ).
t
teT
teT
t
In
This is
1. CONCEPT OF GENERALIZED UNION AND INTERSECTION
53
If Τ = { 1 , 2 , . . . , « } , then
(7)
Π At = AlnA2n...
nAn
teT
If Τ = Jf, where Jf is the set of natural numbers, then the following notation is used :
00
(8)
Π
A
n
instead of
η= 1
Π
AH.
ne^y
Examples
F o r the indexed family (At)teT
numbers, as defined under (I),
of subsets of the set Jf
of natural
= { 6 , 7 , ...} = ^ - { 1 , 2 , 3 , 4 , 5 } .
Π At = A1nA2nA3nA^nA5
fer
For the indexed family (sequence) (An)nçJr
we have
of sets as defined in (II)
n= l
This is true because the number 1 is in each of the sets An9 and n o
other real number is in each An for η eJf. F o r the indexed family
(At)teT
as defined in (III) the following equation holds
C\At
= {0},
teT
since the number 0 is in each of the sets At for t e T9 and no other
real number is in each set At,
teT.
Let (At)t€T
be any indexed family of subsets of X. It follows from
the definition of the intersection of the sets At, t e T, that an element χ
of X is not in (Π At if and only if the condition that χ is in each set At
teT
x
) If Τ = { 1 , 2 ,
/ / } , then the indexed family of sets (At)teT = (Ai9 A2t ...
A„). The generalized intersection Ο At is then the intersection AinA2n
... nAn
teT
in the sense of the definition adopted in Chapter I, Section 3. More generally, if Τ
is a finite set, then Ο At is the intersection, in the sense of the definition given in
teT
Chapter I, Section 3, of a finite number of sets which are in the set of the values of
the function (At)teT. The concept of generalized intersection as introduced in this
section, thus generalizes the concept of intersection, as defined in Chapter I, Section 3, and includes the case when the set Τ is infinite.
54
IV. GENERALIZED UNIONS AND INTERSECTIONS OF SETS
for teT is not satisfied, that is, if there is a t e Τ such that χ φ At.
This is written in symbols as follows:
(9)
(χφΓ)Αί)ο\^(χφΑί).
teT
teT
The properties of generalized unions and intersections will be discussed in the next section.
2. The properties of generalized unions and intersections of sets
The following theorem is a generalization of Theorems 2.2 (7), (8),
(9), and 3.2 (7), (8), (9), Chapter I.
2.1. For any indexed family
of sets
(At)teT:
for every t e Τ', At a (J At and Ο At c
(1)
teT
At\
teT
(2)
if At a A for every t e T, then U At a A;
(3)
if A a At for every t e T, then A <= Π
teT
At.
teT
Condition (1) follows from the definitions of generalized union and
generalized intersection. For if xeAt,
where teT,
then χ is in at
least one summand of the union U At9 and thus also is in that union.
teT
Likewise, if χ e Π At,
then χ is in every set At for teT.
T o prove
teT
(2) assume that At c A for every teT.
If χ e\J
At,
then χ e At
teT
for some t e T. But by assumption At <= A, and hence χ e A.
Conse-
quently, U At cz A. T o prove (3) note that if A a At for every / e T,
teT
A c that
Ç\ χAt.e At for every teT,
meansthe
that
x e Ο Axt. e Hence
then
condition
A implies
teT
which
teT
The following theorems generalizes Theorems 2.2 (10) and 3.2 (10),
Chapter I.
2.2. For any indexed families of sets (At)t€T
t
t
t
and (Bt)teT:
t
every t e T, then teT
(J A c teT
| J B and Π
teT A c= Π
teT B .
if At c
Btfor
2. PROPERTIES OF GENERALIZED UNIONS AND INTERSECTIONS OF SETS
55
Assume that At c Bt for every t e T. If x e (J At> then, for some.
teT
teT,
xeAt.
Since At cz Bt, hence xeBt,
i.e., χ is in at least one
summand of the union [J Bt, so that χ is in that union. This proves
teT
The second inclusion is proved in a
that [J At cz (J Bt.
teT
similar
teT
manner.
The next theorem that follows establishes relationships between
the operations of union and intersection of sets and the generalized
operations.
2.3. For any set A and for any indexed family of sets
(At)teT:
(4)
AuU
At = U
teT
(5)
A, = U
An\J
teT
(6)
ΛυΠ
(AnAt),
teT
A,
=
teT
(7)
(AuAt),
teT
Π
(AKJA,),
teT
A, =
ΑηΠ
teT
Π(ΑηΑ,).
teT
Assume that χ is in the set on the left-hand side of equation (4). Thus
x e A or χ is an element of at least one of the sets At for t e T. Hence
χ is an element of at least one of the sets A uAt, and so χ is in the set
on the right-hand side of equation (4). Conversely, if χ is in the set
on the right-hand side of equation (4), then xeA uAt for some t e T,
so that xeA
or χ e A . If xeA,
then obviously Λ: is in a set on the
t
left-hand side of equation (4). If χ e At,
then χ e[JteT At9
and thus χ
is also in a set on the left-hand side of equation (4). Equation (4) has
thus been proved. The similar proofs of the remaining equations are
left to the reader. Equations (4)-(7) might be called the laws of distributivity. T w o of them, namely (4) and (7), can be generalized thus :
2.4. For any indexed families of sets (At)teT
and (Bt)teT
the following distributivity laws hold:
(8)
UAtuUBt=
teT
(9)
teT
U(AtuBt),
teT
Γ\Α,ηΓ\Β,=
teT
teT
Γ\(Α,ηΒ,).
teT
IV.
56
GENERALIZED UNIONS A N D INTERSECTIONS OF SETS
The proofs are very simple. Equation ( 8 ) will be proved by way of
example. If χ is in the set on the left-hand side of equation (8), then it is in
at least one of the sets
If χ is in the former of these
two sets, then χ e At for some / e T9 and hence χ e At uBt. Likewise,
if χ is in the other generalized union, then χ e Bt for some / e T, and
hence χ e At uBt. Thus in any case Λ: is in a set At uBt, and hence χ
is in
AtuBt).
Conversely, if χ is in the union on the right-hand
side of equation (8), then χ e At uBt for some t e T, and hence χ e A,
or χ G Β, for some t e T. We accordingly conclude that χ e
or χ
Bt. Thus χ is in the set on the left-hand side of equation (8).
Laws (5) and (6) cannot be generalized in the same way as laws (4)
and (7). Only a weaker form of these laws can be proved, namely
2.5. For any indexed families of sets (At)teT
and
(Bt)teT
that,
for
(10)
(11)
Assume that
It follows
Hence χ e At and χ e Bt.
some
/ e 7\
and
Consequently,
Thus χ is in the set on the right-hand side of inclusion (10).
This proves (10). Assume now that χ is in the set on the left-hand side
of inclusion (11). Hence
Accordingly
we
conclude that χ is in every set At (t e T) or that χ is in every set Bt (t e T).
In either case χ is in every set At uBt {t e T), and thus
The proof of 2.5 is thus complete.
Consider now the following two indexed families of sets:
(where Jf is the set of all natural numbers) such that
and
is the set of
where
all real numbers.
Note that for every η EJV the sets An and Bn are disjoint, so that
2. PROPERTIES OF GENERALIZED UNIONS A N D INTERSECTIONS OF SETS
Hence
57
At the same time,
Thus
and
Χ > i j φ ο. This example shows that inclusion (10) may not be replaced by an equation. The sets on the left and on the right-hand side
of inclusion (10) may prove to be unequal to one another.
N o t e further that for these families of set:
and
Hence
same time AnuBn
= 0t for every natural number n, and accordingly
At
the
This example shows that the sets on the right and
on the left-hand side of inclusion (11) may prove to be unequal to one
another, and hence the inclusion symbol in (11) may not be replaced
by the equality symbol.
The following is a generalization of Theorem 4.2, Chapter I.
2.6. For any indexed family
of sets (At)teT
and for any set A:
(12)
(13)
Equations (12) and (13) are the generalized
difference). T o prove (12) note that
and
De Morgan laws (for
At if and only if χ e A
It follows from (5) in Section 1 that this condition is
equivalent to the condition that χ e A and χ $ At for every t e T, i.e.,
χ e A—At for every t e T. This condition is in turn equivalent to the
condition that χ e
A—At).
We have thus shown that
if and only if χ
which proves equation (12). Equation
(13) is proved similarly. The condition χ e A
to the condition χ e A and χ $
is equivalent
, i.e. (see (9) in Section 1) to
the condition χ G A and there is a / e Τ such that χφΑί.
In other
words, χ G A, and χ is not in at least one of the set At, which is equi-
TV. GENERALIZED UNIONS AND INTERSECTIONS OF SETS
58
valent t o the statement that χ is in at least one of the sets A—At.
if and only if xe{J(A-At),
conclude that xeA-C]At
teT
We
which
teT
proves equation (13).
Theorem 2.6 easily yields the following:
2.7. For any indexed family (At)teT
of subsets of a space X:
(14)
At = Π
X-U
teT
(15)
(X~At),
teT
X-r\A,=
\J(X-A,),
teT
teT
so that (see Chapter I, Section 5)
-U A =η - A ,
(16)
t
teT
teT
-ΓΗ,= υ - Λ ·
(17)
(
teT
teT
Equations (16) and (17) are generalized forms of equations (12)
and (13) in Chapter I, Section 5, and are called the generalized De
Morgan laws. They state that the complement of a generalized union
equals the generalized intersection of the complements, and that the
complement of a generalized intersection of sets equals the generalized
union of the complements. The D e Morgan laws are of special importance in applications.
Exercises
00
CO
1. Find [J An and f^] A„ when
η=L
n=i
(a)
An = {xe@:
(b)
An = {xe@\
(c)
An = {xe&\
(d)
An = {xe@:(\
(e)
CO
-\]n
1//?},
<x<
< ·*<
l//f},
1//ι< χ < 2//!},
n
+ \lti)
<x<
3},
An = {x e & : « < A- < Λ + 1 } .
2. Find Π - { A R E Λ : |A| <
M= 1
3. Let
- {xe0t\
χ = sin/}, te0t.
Find (J
and PMJ. Likewise, find
teT
teT
U At and Π Λ , when T= ^ - { 0 } and Λ = {xe$t:
xt < 1}.
/er
teT
EXERCISES
59
4. Let X be any space, and Ο φ V ^ X. Let Η be the family of all subsets of X.
For every Ue Ü, let Au = Un V. Find U Av and Π
l/eit
UeU
5. Prove that
Αυ.
Give an example to show that the symbol <=• cannot be replaced by the symbol = .
6. Prove that
Give an example to show that the symbol <= cannot be replaced by the symbol = .
7. Prove that
···
8. Show that if Αι z> A2
(An)nejV and (Βη)ηε^
are descending
An 3 ... and Bt ZD B2 3 ... z> Bn 3
sequences of sets, then
i.e., if
9. Let (y4„)ne./r be an ascending
Take Ci = Av and C„ = Αη-Αη_γ
sequence of sets, i.e., /4„
Λ π +1 for every
for // = 2, 3, ... Prove that
neJ^.
for every pair of natural numbers n, m such that η φ m, and that
10. Given an indexed family of sets (Αη)η^,
η e Jf. Prove that
Take also d
take Bn = Axv
= Λ and C„ = Λ π —C4„u ... vA„_i)
...vAn
for every
for /i = 2, 3, ... Prove that
for every pair of natural numbers w, m such that η φ m, and that
CHAPTER V
CARTESIAN P R O D U C T S O F S E T S . R E L A T I O N S .
F U N C T I O N S AS R E L A T I O N S
1. Cartesian products
Given any two objects a, b we can form an ordered pair with
antecedent a and successor b, to be denoted by (a, b ) . An ordered
pair (a, b) will be held to differ from an ordered pair (b, a) if α φ b.
Generally speaking, ordered pairs (a, b) and (c, d) are held to be equal
to one another if and only if their antecedents and successors are
equal pairwise, i.e., if and only if a = c and b = d. This convention
will be written in symbols t h u s :
(1)
((a, b) = (c,d))o(a
= CAb = d).
The concept of ordered pair can be introduced in various ways
so as t o satisfy condition ( 1 ) ; for instance, we may adopt the definition
(2)
(a,b)
=
{{a},{a,b}\
The set of all ordered pairs (x, y) such that χ eX and y e Y is called
the Cartesian product of the sets X and Y and denoted by Χ χ Y. Cartesian products of sets occur very often in mathematics. The Cartesian
multiplication of sets is one of the elementary operations performed
on sets. The set # of all complex numbers (the set of points on a plane),
is an example of a Cartesian product; for since a complex number
(a point on a plane) is an ordered pair of real numbers, # is &x 0t,
where 01 is the set of all real numbers.
It can easily be demonstrated that
1.1. If X is a set of η elements, and if Y is a set of m elements, then
XxY is a set of nm elements (ordered pairs (x, y), where χ e X, y e Y).
*) This définition of an ordered pair was given by K. Kuratowski in 1921.
2. BINARY RELATIONS
61
Let m be any natural number, and let Y = {bl9 . . . , 6 m } . If η = 1
and X = {a1}9 then I x Y = { ( ^ , 6 j ) , ..., (α Λ , Z?m)}, and thus consists
of m = 1 · m elements. Assume now that η is a natural number such
that if X has η elements, then Χ χ Y has «m elements. Consider a set
Ζ = { « ! , . . . , tf„, α η + 1} and form the product Χ χ y. Divide the elements
of that product into two sets such that one contains all the pairs (ai9 bj),
where 1
/ ; n, 1 j
m, and the other, all the pairs
(an+l9bj)9
where 1 < j < m . The former set is the Cartesian product of the sets
{al9
an} χ Y. It has, by assumption, nm elements. The latter set
has m elements. Since those sets are disjoint and since their union is
XxY, hence XxY has nm + m = (n+\)m
elements. By applying the
principle of induction we conclude that, for every natural number η
if X has η elements, then XxY has nm elements (ordered pairs). As
m is any natural number, Theorem 1.1 is proved.
2. Binary relations
It was stated in Chapter I, Section 5, that the concept of a subset
of any fixed space X may be identified with the concept of a property
of the elements of that space, i.e., a subset A cz X is identified with
a property WA such that, for every χ eX:
WA{x)
if and only if χ e A.
In this connection the convention is that properties are symbolized
in the same way as sets are, and that the formulations χ is an element
of A and χ has a property A mean the same thing. Under this convention
xeA
and
A(x)
are read either χ is an element of Α ΟΥ χ has a property A.
In mathematics we often consider various binary relations. They
are properties of ordered pairs, that is, properties of elements of a Cartesian product. Thus binary relations are subsets of a Cartesian product.
This will be explained by examples.
We say, for instance, that the relation of divisibility holds between
the natural numbers 2 and 6, or 3 and 15, or 7 and 14. Thus the rela-
62
V. CARTESIAN PRODUCTS OF SETS. RELATIONS
tion of divisibility is a property which is an attribute of ordered pairs
of the natural numbers
(2,6),
(3,15),
(7,14),
...
and of every ordered pair (x, y) of natural numbers such that Λ is a divisor of y, and is not an attribute of any other ordered pair of natural
numbers. Hence this relation is a property of elements of the Cartesian
p r o d u c t s X e / F . Since the properties of elements of the p r o d u c t s xjf
arc identified with subsets of that product, the relation under consideration is a subset of Jf' xjf', namely that subset which contains
an ordered pair (x, y) of natural numbers if and only if χ is a divisor
of y. Let 0t be the set of all real numbers, and let 91 be the family of
all subsets of 0i. The relation of elementhood holds between the
number y2 and the set j l , γ2, j / 3 ] , i.e., } / 2 e { l , | / 2 , \ 3 } . In
general, this relation holds between a real number χ and a subset A
of 0i if and only if xeA.
Thus the relation under consideration is
a property of ordered pairs (x, A), where x e l and A e9l, and is
an attribute of such a pair if and only if
xeA.
N o w let X be any non-empty set, and let 91 be the family of all
subsets of X. The inclusion relation holds between a subset A of X and
a subset Β of X if A cz B. Hence this relation is a property of ordered
pairs (A, B), where A e 91 and Be 91, such that it is an attribute of
such a pair if and only if A cz B. Thus it is a subset of the product
91x91, namely the subset consisting of pairs (A, B) such that A cz B.
In general, the subsets of a product XxY, where X and Y are any
sets, are called binary relations in the product XxY.
If a relation ρ
is a subset of a product XxX, then instead of saying that ρ is a binary
relation in the product XxX we shall often say that ρ is a binary relation on X.
If ρ is a binary relation in Χ χ Y, then instead of (x, y) e ρ we shall
also write χ ρ y, t o be read: χ bears (or is in) the relation ρ to y.
The set of antecedents of ordered pairs (x, y) that are in a relation ρ is called the domain of ρ. It follows from this definition that
an element x e X is in the domain of a relation ρ cz XxY if and only
if there is an y e Y such that χ ρ y. Thus the domain of a relation
ρ cz Xx Y is the set of such elements χ of a set X for which there is
63
2. BINARY RELATIONS
an y e Y such that χ ρ y. Let the domain of a relation ρ be denoted
by @(ρ)· We have
x
(1)
(xe@(Q))o\f( ey)>
N
(2)
yeY
0 ( e ) = {xeX:
/
yeY
\ (xQy)}.
Example
Let X= { 1 , 2 , 3 , 4 , 5 } , Y = {{1, 2 } , {1, 4 } } , and let ρ be the
elementhood relation in the product XxY.
Then Q)(o) = { 1 , 2 , 4 } .
The elements 3 and 5 are not in the domain of ρ, as they are not elements of any of the sets { 1 , 2 } , { 1 , 4 } which are in Y.
The set of successors of ordered pairs (x,y) in ρ is called the
codomain of the relation ρ cz XxY.
It follows from this definition
that an element y e Y is in the codomain of ρ iÇ and only if there is
an x eX such that χ ρ y. Thus the codomain of ρ c Xx Y is the set
of those elements of Y for which there is an χ eX such that χ ρ y.
Let the codomain of a relation ρ be denoted by ^ * ( ρ ) . We have
(3)
(ye2*(Q))o\J{xQy),
xeX
(4)
Example
9*(Q) = {yeY:
xeX
Vixgy)}.
Let X = { 2 , 3 } and let Y = Jf, where Jf is the set of all natural
numbers. Let ρ cz XxJf
be the relation of divisibility, so that
χργ
if and only if Λ: is a divisor of y. Then £2* (ρ) is the set of those natural
numbers which are divisible by 2 or by 3.
Binary relations on the set M of all real numbers can be illustrated
geometrically.
A diagram of a relation ρ c 0tx 0t includes those points
(x,y)
of the plane for which χ ρ y, i.e., (xf y) e ρ. The diagram of the domain
is then obtained by projecting the diagram of ρ onto the axis of abscissae,
and that of the codomain, by projecting the diagram of ρ onto the
axis of ordinates.
Fig. 8 shows a diagram of a relation ρ c 0ί χ 0t which holds between
2
real numbers χ and y if and only if x = y. The domain of that relation, i.e., the projection of its diagram onto the axis of abscissae, is
64
V. CARTESIAN PRODUCTS OF SETS. RELATIONS
the set of all real numbers, and its codomain, i.e., the projection of
its diagram onto the axis of ordinates, is the set of non-negative real
numbers.
The following theorem results easily from Theorem 1.1 and Theorem
2.1 of Chapter TT:
4
Fig. 8
2 . 1 . If X is a set of η elements, and if Y is a set of m elements, then
nm
XxY.
there are 2 binary relations in the product
By 1 . 1 , the set XxY has nm elements. The number of all binary
relations in XxY equals the number of all subsets of XxY,
which,
nm
by Theorem 2 . 1 in Chapter II, equals 2 .
3. Propositional functions of two variables
Given a Cartesian product Χ χ Y of two non-empty sets X and Y,
the formula φ(χ, y), in which two variables, χ and y, occur and which
becomes a proposition if a name of any element of X is substituted
for x, and a name of any element of Y is substituted for y, is called
a propositional function (or a predicate) of two variables, χ and y,
ranging respectively o v e r h a n d Y. T o indicate that χ and y in φ(χ, y)
have X and Y as their respective ranges we shall write
(1)
<p{x,y\
xeX,
yeY.
3. PROPOSITIONAL FUNCTIONS OF TWO VARIABLES
65
A propositional function of two variables, χ and y, which range
respectively over X and Y, may be treated as a propositional function
of one variable, ζ = (x, y), ranging over the Cartesian product XxY.
Thus a propositional function of two variables that range, respectively,
over spaces X and Y, and a propositional function of one variable
ranging over Χ χ Y are one and the same thing.
Example
Let X = Υ = 0t, where M is the set of all real
formula
(2)
x < y,
x e 0t,
numbers.
The
y e @t,
is an example of a propositional function of two variables, χ and y,
which range over 0t and 01.
Consider any propositional function φ(χ, y), xeX,
ye Y. We
say that an element (a, b) of the product Χ χ Y satisfies that propositional
function if the proposition φ(α, b) is true. F o r instance, the ordered
pair (2, 5) satisfies the propositional function (2) as the proposition
2 < 5 is true.
The set of those (x, y) e XxY for which the propositional function φ(χ, y) becomes a true proposition, i.e., the set of those (x, y)
which are in XxY and which satisfy φ will be denoted by
(3)
{(x,y)eXxY:
<p(x,y)}.
It follows from the definition of the set (3) that an element (a, b) of
the product Xx Y is in (3) if and only if the proposition φ(α, b) is true.
This is written in symbols as follows: for every {a, b)
eXxY,
(4)
((a, *) e {(*, y) e Xx Y: <p(x, y)}) ο φ(α9 b).
Thus, every propositional function φ(χ9 y), xeX,
y e Y9 determines a subset {(x9 y) e XxY: <p(x9 y)} of the Cartesian product
XxY,
and hence a relation ρ in XxY.
Accordingly the following
holds for every ordered pair (a, b)
eXxY:
(5)
α ρ b <=> φ(α, b),
i.e., the relation ρ holds between a and b if and only if (a, b) satisfies the propositional function <p(x, y).
66
V. CARTESIAN PRODUCTS OF SETS. RELATIONS
4. Reflexive, irreflexive, symmetric, asymmetric, antisymmetric, and transitive relations
A binary relation ρ cz XxX
is called reflexive if the condition
(1)
χρχ
is satisfied for every χ
eX.
Examples
The identity relation on every set ΧΦ Ο is reflexive, since for every
χ Ε X we have χ = χ. The divisibility relation on the set of natural
numbers is reflexive, since, for every natural number χ, Λ: is a divisor of x. The relation - ("less than or equal to") on the set of real
numbers is reflexive, since, for every real number χ, χ < χ.
A relation ρ cz XxX is called irreflexive if χ ρ χ does not hold
for any χ eX, i.e., if
(2)
~ (χ ρ x)
for every χ
eX.
Examples
The relation "less t h a n " on the set of real numbers is irreflexive,
since the condition χ < χ is not satisfied for any real number x. Likewise, the relation of fatherhood on the set of human beings is irreflexive, since no h u m a n being is his own father.
A relation ρ czXxX
is called symmetric if the condition
χργ
implies y ρ χ for all x,yeX.
In the symbolic notation:
(3)
χ ρ y => y ρ χ
for all χ, y Ε Χ.
Examples
The identity relation on every set Χ Φ Ο is symmetric, since the condition χ = y implies y = χ. The relation of parallelism between straight
lines on the set of all straight lines on the plane 0t χ 01 is symmetric,
since the condition L is parallel to Q implies Q is parallel to L for all
straight lines L and Q.
A relation ρ cz Χ χ X is called asymmetric if the condition χ ρ y implies
~y ρ χ, i.e., if the conditions χργ and y ρ χ are mutually exclusive
for every two elements x, y of X. In symbols:
(4)
χ ρ y => ~(y ρ χ)
for all x, y Ε Χ.
4. RELATIONS
67
Examples
The relation "greater t h a n " on the set of real numbers is asymmetric,
since the conditions χ > y and y > χ are mutually exclusive for every
pair x, y of real numbers. The relation ρ on the set of natural numbers
which holds between χ and y if and only if χ = 2y is asymmetric, since
the conditions χ = 2y and y = 2x are mutually exclusive for every pair
x, y of natural numbers.
A relation ρ cz XxX is called antisymmetric if the conditions χ ρ y
and γρχ
imply χ = y for every pair x,y of elements of X. In the
symbolic notation :
(5)
(χ ρ y Ay ρ χ) => χ = y
for all x, y e Χ.
Examples
The relation < on the set of real numbers is antisymmetric, since the
conditions χ < y and y
χ imply χ = y for every pair x, y of real
numbers. The inclusion relation c on any family of sets is antisymmetric,
since the conditions A cz Β and Β cz A imply A = 2? for every pair A, Β
of sets which are in that family.
A relation ρ cz XxX is called transitive if the conditions χ ρ y and
y ρ ζ imply Λ: ρ ζ for every three elements x, y, ζ of X. In symbols:
(6)
(x ρ y Ay ρ ζ) => χ ρ ζ
for all χ, y, ζ e Χ.
Examples
The divisibility relation on the set of natural numbers is transitive,
since if χ is a divisor of y and y is a divisor of z, then χ is a divisor of
z. The inclusion relation cz on any family of sets is transitive, since
if A cz Β and Β cz C, then A cz C for every three sets A, B, C which
are in that family.
The following theorem establishes a relationship between those relations which are both irreflexive and transitive and those which are asymmetric.
4.1. Any relation ρ cz XxX
asymmetric.
which is both irreflexive and transitive
is
Assume that a relation ρ c l x l i s both irreflexive and transitive.
Assume further that χ ρ y and y ρ χ for a pair x, y of elements of X.
68
V. CARTESIAN PRODUCTS OF SETS. RELATIONS
It follows from the transitivity of ρ that these conditions imply χ ρ x,
which is impossible, since ρ is asymmetric. Hence χ ρ y and y ρ χ are
mutually exclusive, which proves that ρ is asymmetric.
5. Functions as relations
Let X and Y be any sets. If a binary relation ρ <= Xx Y satisfies the
following condition :
(1)
for every χ eX,
there is exactly one yeY
such that χ ρ y,
l
then ρ is called a function ). It associates with every element Λ: of A"
exactly one element y of F, and thus is a function (mapping, transformation) which maps X into Y in accordance with the definition
adopted in Chapter III, Section 1.
Condition (1) can be replaced by the following two conditions:
(2)
for every χ eX, there is an y e Y such that χ ρ y,
(3)
for every x e X and for all yx, y2 e Y, if χ ρ yi and χ ρ y2,
yχ =
then
y2.
Condition (2) provides that every element χ of X bears the relation ρ
to an element of 7, and condition (3) provides that every element χ of
X bears the relation ρ to at most one element of Y. Taken together
these conditions state the same what condition (1) does. Condition (2)
and (3) can be written in symbols as follows:
(2')
/\\/(xoy),
xeX
(3')
yeY
Λ. Λ Λ Ο Q y ι
xeX y yeY
Α Χ
Q y 2) => y\ = y 2)·
y2eY
In other words, condition (2) states that the whole set X is the
domain of the relation ρ, while condition (3) states that every element
of the domain of the relation ρ bears that relation t o only one element
of the set Y. The unique y to which χ bears the relation ρ, i.e., such
that χ ρ y, will be denoted by ρ(χ) and called the value of the function ρ
1
) This rigorous definition of a function in terms of set-theoretic concepts is
due to G. Peano (1911).
EXERCISES
69
for the argument χ. The formula y = ρ(χ) thus states the same what
χ ρ y does. The concept of the domain of a relation then coincides with
that of the domain (set of arguments) of a function, and the concept of
the codomain of a relation then coincides with that of the codomain
(set of values) of a function.
Example
Let Jf be the set of all natural numbers and let & be the set of all
prime numbers. Consider now the relation ρ ο ζ ^ χ { 0 , 1}, defined as
follows :
\(x, 0) e ρ if and only if χ φ0>,
(4)
for every χ eJf\
,
,
^
w
J
1Λ
%c
\(x, 1) G ρ if and only if χ e0>.
The relation ρ satisfies condition (1), for let χ eJf\ if χ e&>, i.e., if χ
is a prime number, then by (4) χ ρ 1, but χ does not bear the relation
ρ to 0; if, on the contrary, χ $0>, i.e., if χ is not a prime number, then
by (4) Λ; ρ 0, but χ does not bear the relation ρ to 1. Hence we conclude
that ρ is a function. It associates with every prime number the number
1, and with every number which is not prime, the number 0. Thus we
have
Let now / : X -> Y be any function that maps X into Y. fis a relation
in XxY, namely such that xfy, i.e., (x, y) e f, if and only if y = f(x).
That relation obviously satisfies condition (1).
Exercises
1. Prove that
Ax(BuC)
=
(AxB)v(AxC),
(BvC)xA
=
(BxA)v(CxA),
Ax(BnC)
=
(AxB)n(AxC),
(BnC)xA
=
(BxA)n(CxA),
Ax(B-C)
=
(AxB)-(AxC),
(B-C)xA
If €Φ Ο, then
A^BoAxC^BxC,
=
(BxA)-(CxA).
and
A <=• Bo
Cx A c C x B.
V. CARTESIAN PRODUCTS OF SETS. RELATIONS
70
2. Determine which of the relations defined below are: reflexive, irreflexive, symmetric, asymmetric, antisymmetric, transitive:
for χ, y e
(a)
χ η
(b)
χ ρy
yoy\x
ol\x-\-y
(c)
χ ρyο
\χ\
(d)
χ οy ο
.ν Ι-vf-1 Ί = 1
for χ,ye
31,
(e)
χ ηyο
χ -\- y = 1
for χ, ye
31,
for x, y E Jf,
for χ, y Ε 01,
< \y\
> 2
for x,
sgn χ < sgn y
for χ, ye
(f)
χ ρ yox+y
(g)
x ρyο
Ν,
yE@,
31.
3. Determine which of the properties specified in Exercise 2 are true of the relation between sets on a plane (relation in the set 2 ^ * ^ ) , defined as follows:
Α η Bo
AnB Φ Ο
and
d{A)<d{B),
where d(X) denotes the diameter of the set X, i.e., the radius of the least circle that
contains X.
4. Find out which of the properties specified in Exercise 2 are true of the relation ρ defined on the set {1, 2, 3, 4, 5} by the following table (where the symbol +
at the intersection of the m-th column and the Λ-th row indicates that (m,n)e ρ):
1
2
3 !4
5
:
1
+ +
2
+ +
+
;
11
+
3
—
+ +
4
5
+
+
-1-
5. Let A 0 be a fixed real number, x 0 > 0, and let ρ, ρ' be relations defined as
follows:
Γ
l> = {(x,y)e3t*x3t\y
= -\ ΧΑΧ
¥
ρ' = {(x,y)ei% x3t:y
< x0}v{(x9y)e3t+x3t'.y
- | / j c } V{(X, y) Ε 01+ x 0t : y =
= \/XAX>
χ0},
-]/*},
1
where 01 is the set of all positive real numbers. Which of these relations is a function? Draw diagrams of the relations ρ and ρ'.
6. Draw diagrams of those subsets of 0t x0t which determine the relations:
Q••={(x,y)z3tx3t:x
< >·},
ρ = {(x,y)e3txâÎ:xï
y},
EXERCISES
Q = {(x,y)eâlxâl:xï
71
yvy
- 2}.
The complements of which of these sets are functions?
7. Give an example of a set and a relation defined on it, such that it has exactly
two of the following three properties : reflexivity, symmetry, and transitivity.
8. Determine the number of all reflexive relations and the number of all symmetric
relations that can be formed in a set of η elements.
9. Define a function that maps {xe0t:
where a <
b,ae&,be&.
0 < χ < 1} onto {x e 0t\ a < χ < 6 } ,
10. Let φ(χ,)>) be a propositional function of two variables ranging over the set
01. Let q = {(x, y)e0txM:
φ(χ, y)}. Determine the domain and the codomain of
o. As the domain of a relation in 01 is a projection of the diagram of that relation
onto the axis of the abscissae, define the projection of the set {(x, y) e 0t x 0t : φ(χ, y)}
onto the axis of abscissae. Prove that the projection of the union of two sets equals
the union of their projections and show by an example that the projection of the
intersection of two sets may differ from the intersection of their projections.
CHAPTER VI
G E N E R A L I Z E D P R O D U C T S . m-ARY R E L A T I O N S .
F U N C T I O N S O F SEVERAL VARIABLES.
IMAGES AND INVERSE IMAGES UNDER A FUNCTION
1. Generalized products
be a given indexed family of subsets of a space X. The
Let (At)teT
Cartesian product of the sets in the family (At)teT
is the set of all those
f u n c t i o n s / : Τ -• [J At which satisfy the condition
teT
(1)
f(t)eAt
for e v e r y /
eT.
That set will be denoted by the symbol
(2)
ΡΛ,·
teT
In particular, if, for every t e T, At = A α X, then the product Ρ At
teT
is the set of all the f u n c t i o n s / : Τ -> A. In such a case we shall write
T
(3)
A
instead of
?At.
teT
Examples
Let Τ = 01, where 0t is the set of all real numbers, and let & be
the set of all integers. Let, for every t e T, At = {x e ££: χ > t}. Then
U At = 3£. The product of the sets which are in the family
(At)1eT,
teT
i.e., the set Ρ At, is, by the definition above, the set of all those functeT
tions f\
0t -+
which satisfy the condition / ( / ) G At for every t e 0t.
The set Ρ At contains, for instance, the function / defined as follows:
teT
AO = [0 + 2 , where [t] denotes the greatest integer less than t. This is
true because, for every / G 0t, J_t] + 2 > t, so that / ( / ) G At. If, for every
1. GENERALIZED PRODUCTS
73
/ e St, we set At = 3ί, then the product Ρ A, = &® is the set of all
f u n c t i o n s / : at -> %.
If Τ = Jf, where .yT is the set of all natural numbers, then instead
of (2) we shall write
00
Ρ A„.
(4)
n=l
00
00
By definition, Ρ An is the set of all those functions f\Jf
-> [J An which
n=l
n=l
satisfy the condition f(n) e An for every « e / ; hence it is the set of all
sequences (an)„ejr
such that, for every ne.V,
the n-th term
aneAn.
\ϊ Τ = Jf and A = A for every « e ^ T , then instead of (4) we shall
n
write
(5)
A*
or
A*«.
By definition, A*° is the set of all sequences (an)ne^for every η
such that ane
A
eJf.
Examples
Let 0 4 η ) „ ε /Γ be the indexed family of sets defined by An = {1, . . . , « } .
00
Then the product Ρ An is the set of all the sequences (an)ne^r
,1 = 1
r
for every η eJ ,
ane
such that,
An, which means that an is a natural number not
00
greater than n. Let now An = {0, 1} for every η eJf.
The product Ρ
n=l
An
such that an = 0 or
= {0, 1}*" is the set of all sequences (an)nejr
r
# n = r 1 for every η eJ . For instance, the sequence defined thus:
(0
for odd a z ' s ,
11
for even «'s
is in that product.
If Τ = {1, ..., m}, then instead of (2) we shall write
Ρ A
(6)
n= l
n
or
By definition, the product Axx
Al x ...
... xAm
xAm.
is the set of all functions / :
m
{1,
m}
[J Λ Λ such that /(w) G Λ„ for every « = 1,
m, so that
π= 1
it is the set of all w-term sequences (al9
am) such that ane
An for
74
VI. GENERALIZED PRODUCTS. /77-ARY RELATIONS
every η = 1 , . . . , r a . If Αη = A for every η = 1 ,
m, then instead of
w e
w r t e
m
i
Of course, ^ is the set of all w-term seAxx ... χΛ„»
quences (ax, ..., am) such that û „ e ^ for every η = 1 , ..., m.
Examples
3
The three-dimensional Euclidean space is the Cartesian product ^ ? ,
where 01 is the set of all real numbers. T o put it more generally, the
m
ra-dimensional Euclidean space is the Cartesian product 0t .
Attention must be drawn to the fact that the product Ax xA2 in the
sense of the definition adopted in Chapter V, Section 1, and the generalized product Ax χ A2 in the sense of the definition adopted in this Section
are not identical, since the elements of the former are ordered pairs,
whereas the elements of the latter are two-term sequences, and these
are two different concepts. In practice differentiation between these two
kinds of products is inessential: we can establish a one-to-one correspondence between every ordered pair {ax,a2)
which is an element of
the product AxxA2
in the first sense of the term and the sequence
(aX9a2)
which is an element of the generalized product AxxA2.
This
is why the same symbols have been adopted for an ordered pair and
a two-term sequence, on the one hand, and for the product Αλ χ A2 of
sets and the generalized product of those sets, on the other.
The following theorem is easy to prove:
1.1. Given sets Anfor η = 1,
with the number of elements of
An equal to kn, the Cartesian product Axx ... χ Am is a set of kx · ... · km
elements.
For Α ι χ A 2 the theorem holds by Theorem 1.1, Chapter V. Assume
that the theorem holds for a natural number j 9 where 2 -· j < m. Let
the product Ax χ ... xAjXAj+x
be formed, and let bl9 ...9bk
be all the
elements of Aj+X.
Let the elements of that product be distributed into
kj+x sets in the following manner: the /-th set, where / = 1,
...9kj+]9
contains all the sequences (ax, ..., aj9 Z?f), where ax e Ax, ..., aj e Aj.
Each of these sets contains by definition kx · ... · kj sequences. These
sets are disjoint and the product under consideration forms their union.
Accordingly, the number of sequences in this product is kx · ... · k} · kj+i.
By applying the principle of induction we conclude that Theorem 1.1
is true.
2. A77-ARY RELATIONS
75
2. m-ary relations
Ternary, quaternary, and, generally speaking, m-ary relations play an
important role in mathematics after binary ones. As is the case of binary
relations, which are properties of elements of a Cartesian product Χί χ
xX2, i.e., are subsets of that product, m-ary relations in X1 χ ... xXm
are properties of elements of that product, that is, of m-term sequences
(al9 ..., am) such that a} eX } for j = 1, ..., m , and hence are subsets of
that product. This will be explained by examples.
The formula χ < y < ζ for x , y, z e 01, where 0ί is the set of real
numbers, refers to a ternary relation on the product 0t χ 0ί x 0t =
0P.
This relation holds, for instance, between the numbers 1, | / 2 , 3 , or
— 1, 0, j / 3 . Thus this relation is a property of three-term sequences of
real numbers
(1,^/2,3),
( - 1 , 0 , j/3),
and, generally speaking, such (ax, a2, a3) that a{ < a2 < a3, but is not
a property of any other three-term sequence of real numbers. Hence this
relation is a property of elements of the product
Since the properties
3
of the product J* are identified with subsets of that product (cf. Chapter
I, Section 5), the relation under consideration is a subset of that product,
namely that subset of it which contains (αί, a2, a3) if and only if
ax < a2 < a3. The formulation "the integers χλ, x2, x3, xA are prime to
4
each other" defines a quaternary relation in the product Jf , where &
is the set of all integers. This relation holds, for instance, between the
numbers 2 , 5 , 7 , 11, and between the numbers 3 , 4 , — 5 , 1 3 . It is a p r o p erty of those, and only those, four-term sequences (al, a2, a3, a 4 ) of
integers for which the numbers αί,α2,α3,
a 4 are prime to one another.
Thus this relation is a subset of the product
namely that subset of
it which consists of those, and only those, sequences (ax ,a2,a3,
aA) in
which all terms form a system of numbers that are prime to each other.
Generally, subsets of the product Xx χ ... x l m , where Xx, ..., Xm are
any sets, are called m-ary relations in that product. If an m-ary relation
m
ρ is a subset of X , then instead of saying that ρ is an m-ary relation
m
in X we often say that ρ is an m-ary relation on X.
If ρ is an m-ary relation in Xx χ ... x l m , then instead of writing
76
VI. GENERALIZED PRODUCTS. W-ARY RELATIONS
(xl9
xm) e ρ we also write ρ(χί9
x m ) , which is read: the relation
ρ holds between xl9 ..., xm.
In the case of m-ary relations, if m > 2 , then the concept of the z-th
domain, / = 1,
m, is introduced. If ρ cz Xx χ ... x l m , then the i-th
domain of that relation is the set of those xt e Xt for which there are
Xj eXj,j φ i,j=
1, ·.·, m9 such that ρ(χί9
x m ) . Thus, for instance,
the first domain of the relation ρ is the set of those xx e Xx for which
there are x2 e X2, . ·., x m e
such that ρ(χι, ..., x m ) .
The following theorem follows easily from Theorem 1.1 above and
from Theorem 2.1 of Chapter II.
2.1. Given a product XYx ... xXm of sets Xj9 j = 1, ...9m9
of kj
kl
elements each, there are in that product exactly 2 "··••*»» m-ary relations.
By 1.1, there are kx · ... · km elements (sequences of m terms each)
in the product Χλχ ... xXm. It follows from Theorem 2.1, Chapter II,
that the number of subsets of this set, i.e., the number of all w-ary
1
relations in the product under consideration equals 2* '···'*"».
3. Propositional functions of m variables
Given a Cartesian product Xx χ ... xXm of non-empty sets Xj9
j — \9...9m9
a formula φ(χ1, ...,xm),
in which m variables
xl9...9xm
occur and which becomes a proposition if the name of any element of
Xj is substituted for each variable xj9 j = 1, ..., m9 is called a propositional function (or a predicate) of m variables xi9
xm which range,
respectively, over the sets Xl9 ...9Xm. In order to indicate that the varim9 as their respective ranables Xj in <p(xl9 ..., xm) have XJ9 j = 1,
ges we write
(1)
φ(χΐ9
GX\,
xm),
3
Consider the product ^2 , where ^2 is the set of all real numbers.
The formula
(2)
2
2
x +y
2
+z
= \9
x9y9ze@9
is an example of a propositional function of three variables,
x9y9z9
which range over the set 01. Let now 2£ be the set of all integers, and
3. PROPOSITIONAL FUNCTIONS OF m VARIABLES
77
jr
let 0t be the set of all sequences whose terms are real numbers. Consider the product 3£ χ 31^ x & . The expression
(3)
an integer χ is a lower bound of a sequence (an)nejr
y is an upper bound of a sequence
(an)neJr
and an integer
is an example of a propositional function of three variables, x,
(a„)nejr9
J
y, such that x e â \ (an)ne^
e M , and y e %.
Propositional functions of m variables, xl9
xm9 ranging respectively over the sets Xi9 ...9Xm9
may be treated as propositional functions of one variable, ζ = (xX9 ..., xm)9 which ranges over the Cartesian
product Xx χ ... xXm. Thus, a propositional function of m variables
ranging, respectively, over the sets Xl9 ...9 Xm9 and a propositional
function of one variable ranging over the product Xxx ... χ Xm are one
and the same thing.
Consider any propositional function <p(xl9 . . . , x m ) , x1 eXl9
...,xm
eXm. We say that an element (ai9
am) of the product Xxx ... xXm
satisfies that propositional function if the proposition φ(αχ,...,
am) is
true. For instance, the three-term sequence (0, 1,0) e
satisfies the
2
2
2
propositional function (2), since 0 + l + 0 = 1. Likewise, the three1
r
term sequence ( 0 , (l/n)ne^-9
2) e 3? x ffl- χ & satisfies the propositional
function (3), since 0 is a lower bound of the sequence (l/w) N E^R> and 2
is an upper bound of that sequence.
The set of those (xl9
xm) eXxx
... xXm for which the propositional function φ(χχ, ..., xm) becomes a true proposition, that is, the set
of those (JC, , . . . , ; c m ) which satisfy that propositional function, is denoted t h u s :
(4)
{(xl9
. . , x J g I , x
...
xXm:
φ(χί9
xm)}.
It follows from the definition of the set (4) that an element (al9 ...
am) of the product Xx χ ... xXm is in the set (4) if and only if the
proposition φ(αχ,
am) is true. In symbols:
(5)
( ( Ö J , ...,am)e
{(xl9
...,xm)eXxx
... xXm:
φ(χχ,
ο
...,
φ(αχ,
xm)})
am).
Every propositional function φ(χχ, ..., xm), xx e Xx, ..., xm e Xm,
thus determines a subset {(xl9 ..., xm) eXxx
... xXm: φ{χλ, ..., xm)} of
the Cartesian product Xxx ... x J m , and hence a relation ρ in
χ
VI. GENERALIZED PRODUCTS. m-ARY RELATIONS
78
χ ... x l f f l. Accordingly, the following relationship holds for every
element (al9
am) of this product
(6)
ρ(αΐ9
...,am)o
φ{αΐ9
...,ûj-
4. Functions of several variables
m
Let X be any non-empty set. A function / : X -» Y is called a function of m variables ranging over X and with values in Y. A function of
m
m variables has for arguments elements of the product X 9 i.e., ra-term
sequences (xl9 ...,xm)
whose terms are in X. The value of a function
/ o f m variables for an argument (xl9 ..., xm) will be denoted by f{xi 9 ...
... 9 xm).
Examples
z
A function / : 0t -» M (where 0t is the set of all real numbers),
defined by the formula f(x9 y9z) = x+y + z is a function of three variables ranging over 0t9 with values in 01. Now let Θ be the family of
all finite subsets of the szXJf (where Jf is the set of all natural numbers).
For every A e Θ , let \A\ denote the number of the elements of the set
4
+
+
A. A function / : Θ -> £ £ (where & is the set of all non-negative
integers), defined by the f o r m u l a / ( Χ , Ζ , V9 W) = \(YuZ)n(Vu
W)\ is
a function of four variables Υ9 Ζ , V, W9 ranging over the family Θ of
sets, with values in
Functions of m variables are special cases of functions of one variable,
and hence all the concepts and theorems in Chapter III pertaining to
functions of one variable are applicable to functions of m variables.
5. Images and inverse images under a function
Given a function / : X
Y9 let A be a subset of X. The image of the
set A under the function f is the set of the images of all elements of A
for the mapping / , i.e., the set of the values of / for those arguments
which are in A. The image of the set A under the function / i s denoted
b y / Ϊ Λ ) ')·
x
) To avoid ambiguity in notation we assume that Λ φ X.
5. IMAGES AND INVERSE IMAGES UNDER A FUNCTION
79
By definition, an element y of a set Y is in f(A) if and only if it is
the image of an element x e A for the mapping / i.e., if and only if
there is son x e A such that y = f(x). In symbols:
(1)
0 ef(A))
ο y {x eAAy
=f(x)).
xeX
Hence it follows that
(2)
f(A) = {yeY:y(xeAAy=
f(x))}.
xeX
Examples
Let fijV -> {0, 1} be the function that maps the set^/T of all natural
numbers into the set {0, 1} and is defined thus:
{
0
if « is an even number,
1
if η is an odd number.
Let A be the set of all odd natural numbers, then f(A) = {1}. Let Β be
the set of all even natural numbers, theh f(B) = {0}. Let C = { 1 , 2 } ,
t h e n / ( C ) = {0, 1}. Now l e t / : 01 -> 01, where m is the set of all real
2
numbers, be a function defined by the formula f(x) = x . Let A be the
set {j/2, 3 , | } , then f(A) = {2, 9, | } . Let Β be the set of all integers,
2
t h e n / ( 5 ) = {0, 1 , 4 , 9 , . . . , « , . . . } .
For those functions / which map the set of real numbers into the set
of real numbers the images of sets under those functions can be illus-
Fig. 9
80
VI. GENERALIZED PRODUCTS. W-ARY RELATIONS
trated geometrically: the image of a set A is the projection of the diagram that corresponds to the set A onto the axis of ordinates.
Consider, for instance, a function / : 01 -> 01 defined by the formula
f(x) = 2x. Let A = {x e 01: 1
χ
2 ] , t h e n / ( / i ) = {y e 0t\ 2 - [y ·
4 } . Geometrically this case is illustrated in Fig. 9.
We now prove a theorem on the image of the union of two sets.
5.1. The image of the union of two sets under a function f equals the
union of the images of those sets under the function j \ so that the following
equation always holds:
(3)
f(AuB)=f(A)uf(B)
\).
T o prove this theorem assume that y ef(A yjB). Then there is an
x e A uB such that y = f(x). In other words, there is an element χ such
that χ 6 A or x e Β and also y = f{x). If χ e A, then y ef(A),
since
y = f(x). Hence, a fortiori, y ef(A)vf(B).
We reason similarly when
x e B. Then y e f(B) and, a fortiori, y e f(A) u/*(2?). We have thus proved
that
(4)
RAuB)
czf(A)uf(B).
Assume now that y ef(A) vf(B): Thus y e f(A) or y ef(B). If y ef(A),
then there is an x e A such that y = f(x). Since χ e A, we have a fortiori,
x e A u 5 , and since y = f(x), y Ε f(A u 5 ) . If y e f{B), then there is an
x e Β such that y = f(x). Since x e B, hence χ Ε A KJB, and since also
y = f(x), hence y ef(A vB). We have thus proved that
(5)
f(A)uf(B)
cz f(A KJB).
Now (3) follows from (4) and (5).
Theorem 5.1 can be generalized to read:
5.2. Let f: X -> Y and let (At)teT be any indexed family
X. The following equation then holds
of subsets of
/(UA) = UM).
(6)
teT
Assume that y e / ( U
At).
teT
It follows from the définition of the image
teT
x
) Theorem 5.1 follows from the more general Theorem 5.2. Hence the proof
of Theorem 5.1 could be omitted here. It is given here solely for didactic purposes.
81
5. IMAGES AND INVERSE IMAGES UNDER A FUNCTION
that there is an χ e ( J
A
teT
t such that y = f(x). Hence there is an χ such
that it is in at least one set At and such that y = f(x).
Consequently,
for some teT,
We have thus
If follows that y e ( J f(At).
ye f(At).
teT
proved that
/(U^)CUM).
(7)
teT
Assume now that y e (J f(At).
teT
y e f(At).
teT
If so, then there is a / € Τ such that
Hence it follows that there is an χ e At such that y =
Since χ e At, hence χ e{J
At; since at the same time y = f(x),
f(x).
hence
teT
y ef{\J
At). We have thus proved that
teT
(8)
UM)C/(U4).
teT
teT
Now (6) follows from (7) and (8).
The following theorem pertains to the image of the intersection of
two sets.
5.3. The image of the intersection of two sets under a function f is
contained in the intersection of the images of those sets under the function
f so that the following inclusion always holds:
(9)
f(AnB)
<=f(A)nf(B)
l
).
Assume that y e f(A nB). Hence there is an x e A nB such that y
= f(x). In other words, there is an χ such that x e A, x e B, and y
= f(x). Since x e A and y = f(x), hence y e f(A). Since at the same time
x e Β and y = f(x), hence y e f(B). This yields y e f(A) and y e f(B),
that is, y e f(A) nf(B). This proves inclusion (9).
In formula (9), the symbol of inclusion may not be replaced by the
symbol of equality. For instance, l e t / : M
where ^? is the set of
all real numbers, be the function defined by the formula f(x) = \x\. Let
A = {— 1} and let Β = {1}. The intersection of these sets is the empty
set: A nB = O. The image of the empty set is the empty set, hence
0 Theorem 5.3 follows from the more general Theorem 5.5. Hence the proof
of Theorem 5.3 could be omitted here. It is given here solely for didactic purposes.
VI. GENERALIZED PRODUCTS. W-ARY RELATIONS
82
f(A nB) = 0. At the same time we have f(A) = {1} and f(B) = {1},
and hence f(A)nf(B)
= {1}. This example shows that the image of the
intersection of two sets may sometimes not contain the intersection of
the images of those sets, so that, accordingly, the image of the intersection is not necessarily equal to the intersection of the images. In the
example now under consideration fis not a one-to-one function.
5.4. If f: X -> Y is a one-to-one function, then for any subsets A, Β of
X the following equation holds:
(10)
f(AnB)
= f(A)nf(B)
*).
In view of 5.3 it suffices to prove that, under the assumption that /
is a one-to-one function, the inclusion f(A)nf(B)
c f(A nB) holds.
Assume that y e f(A) nf(B). Hence, y e f(A) and y e f(B). Consequently,
there is an xl e A such that y = f(xi), and there is an x2 e Β such that
y = f(x2). Consequently,/(xj) = f(x2)> Since fis a one-to-one function,
this implies that x1 = x2. Hence x{ e A and xx G Β and y = / ( J C 1 ) .
Accordingly, xl eAnB and y = f(xx), so that y e f(A nB), which proves
Theorem 5.4.
Theorem 5.3 can be generalized thus:
5.5. Let f: X -> Y and let (At)tsT be any indexed family of subsets of
X. The following inclusion then holds:
(Π)
/(ilA)cnM).
teT
Assume that y ef(D
At).
teT
It follows from the definition of the
teT
concept of image that there is an χ e Π At such that y = f(x).
Hence
teT
it follows that there is an χ which is an element of each of the sets At
for t e T, and such that y = f(x).
Consequently, for every t e Τ, χ e At
and y = f(x),
for every teT.
so that y e f(At)
Hence y e P|
f(At),
teT
which proves (11).
In (11), the inclusion symbol may not be replaced by the equality
symbol because, as we have shown, the inclusion symbol may not be
J
) Theorem 5.4 follows from the more general Theorem 5.6. Hence the proof
of Theorem 5.4 could be omitted here. It is given here solely for didactic purposes.
5. IMAGES AND INVERSE IMAGES UNDER A FUNCTION
83
replaced by the equality symbol in (9). The following theorem is a generalization of 5.4.
5.6. If f: X -> Y is a one-to-one function, then the following
of subsets of X:
holds for every indexed family (At)teT
(12)
/(Π
A)
=
teT
equation
C\f(A,).
teT
T o prove (12) it suffices to demonstrate that the inclusion C\
f(At)
teT
a / ( P | ^ 4 r ) holds if / is a one-to-one function (as the converse infer
elusion holds by 5.5). Assume that j e f ] f(At).
It follows that y e f(At)
teT
for every t e T. Consequently, for every / e T, there is an xt e At such
that y = f(xt).
Since fis
a one-to-one function and f(xt)
= y for every
/ e Γ, that is, the images of the elements xt are equal, hence all xt must
be equal to each other. Thus, there is an χ such that xt = χ for every
t e T. Consequently, χ is in each of the sets At9
and f(x) = y. Hence
x e Π At
which proves The-
and y =f(x),
so that y ef(D
teT
orem 5.6.
5.7. Iff:
A, Β ofX:
(13)
At),
teT
X -> Y, then the following
f(A)-f{B)
cz
inclusion holds for any subsets
f(A-B).
Assume that yef(A)-f(B).
It follows that yef(A)
and
y$f(B).
Since y ef(A), hence there is an x0 e A such that y = / ( χ 0 ) · It follows from the condition stating that y φf(B) that there is in I no χ
such that x e Β and at the same time y = f(x). In other words, for
any element χ eX, the conditions x e Β and y = f(x) cannot both be
satisfied. Consequently, for any xeX,
if y =f(x),
then χφΒ.
In
particular, since y = f(x0)9
hence x0 φΒ. Thus, x0 eA, χ0φΒ9
and
y = / ( * ο ) · Consequently, x0 eA — B and y = f(x0).
Hence it follows
that y ef(A-B),
which proves (13).
5.8. Iff: X -> Y is a one-to-one function,
holds for any subsets A, Β of X:
(14)
f{A-B)=f{A)-f(B).
then the following
equation
84
VI. GENERALIZED PRODUCTS. W-ARY RELATIONS
To prove (14) it suffices to demonstrate that, under the assumption
that / is a one-to-one function, the following inclusion is satisfied:
(15)
AA-B)
cz
f(A)-f(B).
The converse inclusion is always satisfied by 5.7. Assume that y e f(A — B).
Hence
This means that there is an x0 e A - Β such that y = f(x0).
x0 e A, x0 φ Β, and y = f(x0).
It follows from x0 e A and y = f(x0)
that y e f(A). Suppose that y e f(B). This yields that there is an x e Β
such that y = f(x). As / is a one-to-one function and the conditions
a n
y = f(xo)
d y = f(x) are both satisfied, we conclude that x0 = x.
Yet x0 φ B, and x e B, which yields a contradiction. Hence y φ f(B).
Since also yef(A)9
we have yef(A)—f(B).
This proves inclusion
(15) and Theorem 5.8.
The following theorem will now be proved:
5.9. Let f\ X -+ Y be a function that maps X into Y. If A and Β are
subsets of X such that A cz B, then f(A) cz f(B).
Assume that y e f(A). Thus there is an x e A such that y = f(x).
If x e A and A cz B, then x e B. Since also y = f(x)9 hence y e f{B).
T h u s / 0 4 ) cz f(B).
Let f: X -> Y. The set of the values of the function / i.e., the codomain of the function / was defined in Chapter III, Section 1, as
the set of those elements of Y for which there is an x e l such that
y =f(x).
By the definition of the image of a set under a f u n c t i o n /
the set of values of the function / coincides with the image of X under
/ This is why the symbol f{X) has been adopted for the set of the values
of the function / : X -> Y.
The second important concept connected with that of function,
namely the inverse image, under a function / : X -> Y, of a set included in y, will now be defined.
Let f: X -+ Κ be a function that maps X into Y and let C cz Y. The
set whose elements are those elements of X whose images are in C is
called the inverse image of the set C under the function f It follows
from this definition that χ is an element of the inverse image of C,
for any χ eX, if and only i f / ( x ) e C. The inverse image of C u n d e r /
_ 1
J
is denoted b y / ( C ) ) . The above definition yields:
\) To avoid ambiguity in notation it is assumed that C φ Y.
5. IMAGES AND INVERSE IMAGES UNDER A FUNCTION
(16)
(17)
85
xef-\C)of{x)eC,
f-\C)
=
{xBX:f{x)eC}.
Examples
Let / : M
0ί (where M is the set of all real numbers) be a function which is defined t h u s :
0
for irrational x's,
1
for rational x's.
The set of irrational numbers is the inverse image of the set {0}, a n d
the set of rational numbers is the inverse image of the set {1}. If the
set of all rational numbers is denoted by Q, the following equalities
result:/"UW) = Λ - Q and f-\{\))
= Q. Let now / : » -> St be
a function defined by the formula fix) =
and let C = {y e 0t:
1
y < 2 } . Then
= {* e St: 1 < χ <;; 2} KJ{X e 01: -2 < Λ:
In the case of those functions / which m a p the set of real numbers
into the set of real numbers the inverse images of sets under those
functions can be illustrated geometrically. The inverse image of C is
represented by t h e projection of the diagram of the function which corresponds t o the set C onto the axis of abscissae. F o r instance, for the
function / and the set C in the example last described the formation
of the inverse image is illustrated in Fig. 10.
The following theorem states that theorems analogous t o 5.1, 5.2,
5.4 (without the assumption that / is a one-to-one function), 5.6 (with-
A
r*(c) = AuB
Fig. 10
86
VI. GENERALIZED PRODUCTS. W-ARY RELATIONS
out the assumption that / is a one-to-one function), 5.8 (without the
assumption that / is a one-to-one function), and 5.9 hold for inverse
images.
5.10. Let f: X
Y be a function that maps X into Y. The following
equations hold for any subsets C and D of Y and for any indexed family
of subsets of Y:
(Ct)teT
1
1
(18)
tKCKjD)
=/" (C)u/- (i)),
(19)
/ l u c r u m ,
teT
teT
1
I
I
(20)
/- (CnZ))=/- (C)n/- (£>),
(2D
/- (n
x
ct)
=
rv-HQ,
teT
(22)
teT
f~\C-D)
=f-\C)-f-\D).
If C <= D, then
(23)
f-'{C)
c Z - H D ) .
- 1
Assume that x e / (C u D ) . This means that f(x) e CuD.
This
condition is equivalent to the condition that f(x)eC
or
f(x)eD9
_ 1
-1
which in turn is equivalent to the condition that χ e / ( C ) or χ
e/ (Z>).
This last condition is satisfied if and only if χ ef^i^uf^iD).
This
proves (18).
T o prove (19) note that x e / ^ U
teT
Ct) if and only iff(x)
e U
teT
Q,
i.e., if f{x) is in at least one of the sets Ct. This is equivalent to the
condition that χ is in at least one of the sets /
- 1
( 0 > which in turn
is equivalent to the condition: x e U/"HQ)- This concludes the proof
teT
of (19).
T o prove (20) note that the condition x e /_ 1 ( C n Z ) ) is satisfied
if and only if fix) e CnD, i.e., if fix) e C and fix) e D, which is equivalent to the condition x e / "1 (C) and χ e / "1 (Z>), i.e., x e / "1 (C) η / -1 (Ζ)).
Ct) if and only if fix) e Π
T o prove (21) note that χ ef-^H
teT
Ct,
teT
which means that fix) is in each of the sets C,, for / e T. This occurs
5. IMAGES AND INVERSE IMAGES UNDER A FUNCTION
87
if and only if χ is in each of the sets / " ^ ( C , ) , i.e., if and only if χ
1
€ Π Ζ"
T
hs
i
P
r o v es
(21)·
teT
T o prove (22) note that χ ef-^C-D)
if and only 'ύ fix)
eC-D,
i.e., if and only if f(x)eC
and f(x)$D.
This condition is satisfied
1
l
if and only if χ ef~\C)
and χ φ/- φ) y i-e., if and only if x
ef~ (C)-f-^D).
This concludes the proof of (22).
Assume now that C CZ D. If χ ef-^C),
then f(x) e C. Since C cz D,
_ 1
hence f(x)eD,
and thence it follows that x e / ( Z > ) . This proves
1
that (23) then holds. This completes the proof of Theorem 5.10 ).
The theorem to be proved now relates the concept of image to
that of inverse image.
5.11. Let f: X
Y be a function that maps X into Y. The following equation holds for every subset C of f(X):
(24)
fit
1
(CS) =
C
The following inclusion holds for every subset A of X:
(25)
A
<=f-\M).
If f is a one-to-one function, then, for every A cz X9
(26)
A
1
=/- (/(^)).
T o prove (24) assume that y e / ( / ( 0 ) - It follows from the definition of the image that if this assumption is true then there is an
- 1
_ 1
x e / ( C ) such that y = f(x). But the condition x e / ( C ) is equivalent t o the condition f(x)eC.
Since y = f(x), we conclude that
C
yeC.
We have thus proved that /(/"HO)
C. Assume now that
y e C cz f(X). Since C cz f(X), hence every element of C, in particular
y, is the image of an element of X. Hence there is an x e X such that
-1
y = f(x). Since y eC, hence f(x) e C . It follows that χ e/ (C).
Hence
x
the condition yeC
implies the condition that there is an
xef~ (C)
such that y = f(x). We infer from the definition of the image that
- 1
y e / ( / ( C ) ) . We have thus proved that C cz /(/"HC)), which, taken
together with the inclusion proved above, proves (24).
-1
0 The proofs of equations (18) and (20), which follow from the more general
formulas (19) and (21), could be omitted in the proof of Theorem 5.10. They are
given here solely for didactic purposes.
VI. GENERALIZED PRODUCTS. W-ARY RELATIONS
88
T o prove (25) assume that x e A c X. Accordingly, the image
1
of x is in f(A), i.e., f(x) ef(A). This implies that χ e / " ^ ) ) , which
proves (25).
l
Assume now that / is a one-to-one function. If χ ef~ ^f{A)),
then
it follows from formula (16) that f(x)ef(A).
Consider now the definition of the image. The fact that f(x) ef(A)
means that there is
an x0 Ε A such that f(x) = f(x0).
But, since f is a one-to-one function, hence the condition f(x) = f(x0)
implies that χ = x0.
Since
1
XQEA,
hence χ Ε A . We have thus proved that / ~ ( / ( ^ ) ) c A .
This and (25) yield (26).
The following example shows that if fis not a one-to-one function,
then equation (26) need not hold. Let / : 01 -+ ffl (where 0t is the set
of all real numbers) be a function defined by the formula j\x) = \x\.
1
1
Letalsoy* = ( l } . T h e n / ( ^ ) = {1} a n d / - ^ ) ) = / " ( { l } ) = { - 1 , 1}·
Hence A
^f~\f(A)).
Exercises
1. Prove that if a set Τ has a finite number of elements and At φ Ο for every
t G Γ, then Ρ Λ ^ Ο.
Note. A more general theorem, which does not assume that the set Τ is finite,
needs the axiom of choice in its proof.
2. A function f which maps the set of non-negative real numbers into 0t is defined thus :
(a) f{x) - \x~2n-\\
for In < χ < 2(/Η-1), Λ = 0, 1, ...,
(b) /(JC) - x-2n
for 2n < χ < 2(/i-|-l),/f == 0, 1, ...
l
Find the image f(A) and the inverse image f~ (B)
(a) Λ - U
! * e # : Y + 2rt< * < - f + 2rt},
of the sets:
£={ve^:0<;y<-M,
n=o
(b)/i = U { * e ^ : 2 / / - 1 < JC < 2/i},
η=Ι
5 - {ye
y < .y < y } ·
3. Given a function / : A"-* ^£ and sets A and i?, find the image f(A) and the inverse
-1
i m a g e / (B) for:
00
(a)/(jc) = sin2*,
£ = {ye0t\
= {xe 0t\ x > 0 } ,
- 1 <>><0},
Λ = ( J ( J C G ^ : π// < * < y π + π / / | ,
EXERCISES
2
(b)/(jc) = | x - 4 | , X = 0t,
89
Λ = {x e 0t: 0 < x < 1}, 5 = {ye
0t: 2 <>> < 4 } ,
2
(c)/(x) = | Χ - 2 Λ : | , * = ^ , Λ = { * ε ^ : \x\ < 1}, £ = {>>e ^ : 0 < >> < i } .
4. Given a function/: 01-+ 0ί defined by the formula/(*) = \x+\\
l
Λ = {JCG 01 \ 1 < JC < 2 } , find the sets f(A) and
f~ (f{A)).
6
and a set
4
2
5. Given a function / : 0t-+ 0t defined by the formula f(x) = Χ + Λ : + Λ : + 1 ,
1
find f(0t),f({xe0t:
-2 < χ < 2}) a n d / - « ^ ^ : 0 < JC < 1}).
6. Let
M stand for the set of all polynomials with real coefficients of degree
2
< rt,n > 1 . Let g(f) for fe 0tn\x\ be defined thus: g(f) = f(x + \). Find the inverse image of the set of all polynomials of the degree 0 , the inverse image of the
2
set that consists of the polynomial x -f 1 only, and the image of the set that consists
of all polynomials of a degree < 1.
7. The function fA defined by:
(
1
for*ey4,
0
for Λ: e Χ— A,
is called the characteristic function of the subset A c χ. Prove that
where 0 0 0 = 0 and 0 0 1 = 1 0 0 = 1 0 1 = 1,
8. Prove that a necessary and sufficient condition for a function that maps X
onto F to be a one-to-one function is that the inverse image of every one-element
subset of y be a set of one element (a one-element subset of X).
9. Prove that/(Λ n /
1
(#))
-=f(A)nß.
l
10. Let g = f\A (cf. p. 39). Prove that g~ (B) -
l
Anf (B).
CHAPTER VII
1
EQUIVALENCE RELATIONS
)
1. Definition of equivalence relations. Method of identification
A relation ρ cz XxX is called an equivalence relation on X if ρ is
reflexive, symmetric, and transitive, so that (cf. Chapter V, Section 4)
the following three conditions are satisfied:
(r)
χρχ
for every χ
(s)
χ ρ y => y ρ χ
for every χ, y e Χ,
(0
(χ ργ/\y
for every χ , y, ζ
ρ ζ) => χ ρ z
eX,
eX.
An equivalence relation is usually denoted thus:
Examples
(I) Let 0t be the set of all real numbers and let » be a binary relation on 01 defined t h u s : for any real numbers x9y, the relation χ & y
holds if and only if x—y is an integer. Let J? stand for the set of all
integers. We accordingly have
(1)
χ « y ο x—y e 2£.
The relation « thus defined is an equivalence relation on 0t. This relation is reflexive, as x — x = 0 e
, and hence χ « χ. The relation «
is symmetric, since if χ « y, then x—y e 2t', which yields that y — x
= —(x—y) e 3t 9 and hence y « χ . It is equally easy to show that the
relation « is transitive. Assume that x & y and y « ζ. It follows therefrom that x - j e J and J ; - Z G ^ . Consequently, x - z = (x—y) +
+ (y-z)e&,
so that Λ: « ζ.
(II) Let JS? be the set of all straight lines on the Euclidean plane
A
2
0l .
) The great importance of these relations in the various branches of mathematics
was first indicated by G. Frege (1884).
1. DEFINITION OF EQUIVALENCE RELATIONS
91
Let « be a binary relation on J£\ such that Ρ « Q, for any straight
lines P, Q in
if and only if Ρ is parallel to g . This relation is an
equivalence relation on 5£.
(III) Let S2 be the set of all bound vectors in the plane
i.e., vectors
(p, q), where p, q are any points of the plane, lip = (xx, yt), q = (x2,
then x2-Xi,y2~yi
ea r t ne
y2),
coordinates of the vector (/?, qj. Let « be
the binary relation on S2 which holds between any two vectors (/?, q)
« Ο ι , qù if and only if the coordinates of the vector (pTq) equal the
coordinates of the vector (/?!, qx). It can easily be verified that the relation « thus defined is an equivalence relation on
S2.
(IV) Let X be any non-empty set and let « be the equality relation
on Χ: χ « y if and only if Λ: = j>. This relation also is an equivalence
relation on X.
Let « be any equivalence relation on Χ Φ O. For every element
x e l , let 11*11 denote the set of all those elements y eX which bear the
relation « t o *, i.e., such for which the condition χ « y is satisfied.
Thus by definition
(2)
(3)
IWI =
(y e 11*11) <^> (* « j>)
{yeX:x*y},
for every x,y
eX.
The sets ||*|| for χ eX are called equivalence classes of the relation
«
More exactly, a class ||*|| is called an equivalence class of the
relation « on X determined by χ or represented by x. The set of all equivalence classes of a relation « on ^ is denoted by X/&.
Examples
In example (I) the set of all real numbers y such that *—j> is an
integer is an equivalence class ||*|| for every real number x. For instance,
if m is an integer, then every integer is in the class \\m\\. Hence we infer
that all integers are in one equivalence class of the relation « , and that
n o real number which is not an integer is in that class.
In example (II), every straight line Q parallel to Ρ is in the equivalence class determined by any straight line P, i.e., in the class | | P | | ,
and n o other straight line is in that class. The equivalence classes of
VII. EQUIVALENCE RELATIONS
92
that relation on
i.e., equivalence classes of the relation of parallelism
between straight lines are called directions.
In example (III), every bound vector (pl9
qx) whose coordinates are
respectively equal to the coordinates of a bound vector (p, q) is in the
equivalence class determined by that vector (p,q),
\\(p> tf)ll>
a n (
i
n o
i.e., in the class
other vector is in that class. The equivalence classes
a r
of that relation on S2 e called free vectors.
In example (IV), the equivalence class of the relation » determined
by any element x, i.e., the class \\x\\, has only one element, namely, x.
The following theorem lays down the basic properties of the equivalence classes of any equivalence relation.
1.1. Let « be any equivalence relation on a set Χ Φ Ο. The following
conditions are then satisfied for every x,xl9x2
eX:
(4)
xe\\x\\,
(5)
\\xi\\ = \\X2W if and only if Χ! κ
(6)
//Ikill
x2,
Φ I t a l l » then the classes \\χλ\\ and \\x2\\ are disjoint.
The relation « , being an equivalence relation, is reflexive, and hence
for every x e l w e have χ » χ, so that χ e \\x\\.
Assume now that \\χλ\\ = \\x2\\. This and (4) yield that xx e | | * 2 | | .
In view of (3) and (s) we obtain xx » x2. Conversely, if Xi & x2, then
I l * i 11 = l l * 2 l I- For if χ e \\xx ||, then it follows from (3) that xx « x9
from which we further infer, b y (s), that χ « xx. Also, xt « x29 by
assumption. Since « is a transitive relation, hence χ » x2. By applying
(s) and (3) we obtain that x e | | x 2 | | - Consequently, \\χχ\\ cz \\x2\\. If
χ e | | x 2 | | > then it follows by (3) that x2 « x. At the same time Xi & x2.
c
By (t), Xi « x. By applying (3) we obtain χ e \\χλ\\. Thus, \\x2\\
ll*ill>
which, taken together with the inclusion proved above, yields \\χχ\\
— 11*2II· We have thus proved (5).
T o prove (6) assume that \\Xi\\ and \\x2\\ are not disjoint. There is
then an element χ such that χ e \ \χχ\\ and χ e \ \x2\\. Using (3) we infer
that Xi « * and x2 « χ. This and (s) yield xx « χ and χ « x 2 . By (t),
a n
« x 2 , which, in view of (5), implies that \\Xi\\ = | | x 2 | | d proves (6).
Theorem 1.1 yields the following theorem, called the principle of
identification of equivalent elements:
2. CONSTRUCTION OF INTEGERS
93
1.2. Any equivalence relation « on a set Χ Φ Ο establishes a partition of the set into disjoint non-empty subsets, namely the equivalence
classes of the relation, so that any two elements x,y eX are in the same
equivalence class if and only if χ « y.
When we pass from elements x,y of X to equivalence classes \\x\\,
\\y\\, the equivalence relation « is replaced by the relation of equality.
This method is very often used in mathematics, and it will be called the
1
method of identification of equivalent
elements ).
This method, applied as in example (II), yields a definition of directions, as applied in example (III) yields a definition of free vectors.
Other examples of its application will be given in sections that follow.
2. Application of the method of identification to the construction of integers
Consider the Cartesian product Jf xJf 9 where Jf is the set of all natural numbers. The elements of that product are ordered pairs (m,n),
where m, η are any natural numbers. Let Ä be a binary relation in
Jf xJf defined t h u s :
(1)
( m 2 , ntl) « (m2, n2) if and only if m1+n2
=
m2+n1.
Thus the relation « holds between two ordered pairs (m1,n1)
and
(m2, n2) of natural numbers if and only if the difference between their
successors is the same as that between their antecedents. For instance,
( 1 , 1 ) « ( 2 , 2 ) , ( 3 , 5 ) « (7, 9), ( 8 , 3 ) « ( 6 , 1 ) .
2.1. The relation « as defined by formula (1) is an equivalence relation inJf
xJf.
The relation « is reflexive, since, for every ordered pair (m,n)
a s m + n = m + n.
eJf xJf', (m9n) «
The relation « is symmetric, for if ( / w l 5« , ) « (rn29n2),
then it
follows from Definition (1) that ml+n2
= m2+nl.
Hence
m2-\-n1
= m ! + « 2 , which means that (m2, n2) « (m^, n^.
The relation « is transitive. For assume that (ml9n1)
«
(m29n2)
and that (m2, n2) « (m3, n3). Then the following equations hold in view
*) This method is used very often in mathematics, in particular in forming definitions of new mathematical entities.
V I L EQUIVALENCE RELATIONS
94
of (1): ml + n2 = m2+nl
and m2+n3
= m3+n2.
Hence mL+n2+m2
+
+ n3 = m2+n1+m3+n2,
i.e., (m1 +«3) + (rn2 + n2) = (m3+n1) + (m2 +
= m3+nl.
In view of (1) this equation yields
+ n2), and hence m1+n3
(raj, nx) « (m3, n3). The proof of 2.1 is thus complete.
It follows from Theorem 1.2 that the relation « divides the set
Jf xjf
into disjoint and non-empty equivalence classes, so that two
ordered pairs (mx, nx) and (m2, n2) of natural numbers are in one and
the same class if and only if {mx , nx) » (m2, n2).
The equivalence classes of the relation « in the set^/T xj^ are called
integers.
It can easily be seen that ( 1 , 1) « (2, 2) « ( 3 , 3) « ... The class
| | ( 1 , 1)|| determines the integer 0. The classes | | ( W , J Î ) | | , where m > η
and m = n + k, k = 1 , 2 , . . . , determine positive integers k = 1 , 2 , . . .
The classes | | ( m , n ) | | , where m < η and η = m + k, k = 1 , 2 ,
determine negative integers — k=—1,-2,
...
The operations of addition and multiplication on those numbers are
defined thus :
I K ^ i , Λι)ΙΙ + Ι Ι ( Ή 2 , * 2 ) Ι Ι =
\\(ml9 «OH · \\(m29n2)\\
=
Ι Ι ( ^ ι + ^ 2 , « ι + « 2) Ι Ι ,
\\{mlm2+nln29mln2+n1m2)\\.
It can be proved that the operations so defined do not depend on the
choice of representatives of equivalence classes and that they comply
with all the laws known in the arithmetic of natural numbers ; moreover,
that substraction can always be performed in the domain of integers *).
The definition of integers as formulated above may seem somewhat
unnatural, but if we take into account the fact that integers make it
possible t o determine how much a number exceeds or falls short of
another one, the definition becomes more intuitive. It also has the advantage of making it possible t o construct integers in terms of natural
numbers and set-theoretical concepts.
l
) Proofs of these theorems may be found in: E. Landau, Grundlagen der Analysis, Leipzig 1930.
3. CONSTRUCTION OF RATIONAL NUMBERS
95
3. Application of the method of identification to the construction of
rational numbers
Let Jf be the set of all integers, and let 2£* be the set of all integers
other than 0. Consider now the Cartesian product 3£ x 2Ê* : its elements
are ordered pairs (m, n) of integers such that their successors are other
than 0. Let « be a binary relation in J? χ 2£* defined thus :
(1)
(m1, η χ) « (m2, n2) if and only if m1 n2 = m2 n1.
and
Thus the relation « holds between any ordered pairs (ml9ni)
( m 2 , n2) of integers (n1 Φ 0 and n2 Φ 0) if and only if the ratio of the
antecedents is the same as that of the successors, e.g., ( 3 , 5 ) « (15, 25),
( 1 , 2 ) « ( 7 , 14), ( - 5 , 2 ) « ( 1 5 , - 6 ) .
3.1. The relation « as defined by formula
tion in the set 2t χ Jf*.
(1) is an equivalence rela-
The relation « is reflexive, since, for every ordered pair (m,n)
e Jf χ Jf*, (m, n) « (m,n), as mn = mn.
The relation « is symmetric, for if (ml9n1)
« (m2,n2),
then it
follows from (1) that mxn2 = m2n1. Hence m2n1 = m1n2, which means
that (m2,n2)
«
(ml,n1).
The relation « is transitive, for assume that (mx, nx) « (m2, n2) and
(m2,n2)
« (m3,n3).
Then m1n2 = m2nY
and m2n3 = m3n2.
Hence
m1n2m2n3
= m2nlm3n2,
so that ( m ^ ) (m2n2) = (m3n1) (m2n2).
It
follows that if w 2 ^ 0, then m1n3 = m3nl9 which proves that ( m ^ « : )
« (m3,n3).
If m2 = 0, then both m A = 0 and m 3 = 0. Then, too,
m±n3 = m3nl9 so that im1,nl)
« ( r a 3 , « 3 ) . The proof of 3.1 is thus
finished.
It follows from Theorem 1.2 that the relation « establishes a partition of the set Jf χ Jf * into disjoint non-empty equivalence classes in
such a way that two ordered pairs (mx, nx) and (m2, n2) are in one and
the same class if and only if (mx, nt) « (m2, n2).
The equivalence classes of the relation « in Jf χ Jf * are called rational numbers. The classes | | ( m , « ) | | determine the rational numbers
m/n. Ordered pairs
and (m2,n2)
determine one and the same
class, that is, one and the same rational number, if and only if mtn2
= m2n1. If it is the case, then the rational number mljn1 equals the
rational number
m2/n2.
VII. EQUIVALENCE RELATIONS
96
The operations of addition and multiplication on rational numbers,
i.e., on equivalence classes of the relation « in
x ^ f * , are defined
thus:
" ι ) Ι Ι + Ι Ι θ 2 , «2)11
=
I I ( w 1/ î 2+ w 1/ w 2, , i 1w 2) | | ,
| | 0 * L , > * L ) l l · 1 1 0 * 2 , "2)11
=
110*1^2,
« I / F 2) | | .
It can be proved that the operations thus defined do not depend on the
choice of the representatives of the equivalence classes and that they
comply with all the laws known to hold in the arithmetic of rational
numbers, and also that division by a rational number other than 0,
i.e., other than ||(w, « ) | | , where m — 0, can always be performed in the
domain of rational numbers.
An exposition of the theory of rational numbers defined in the above
manner can be found in Landau's book referred to Section 2.
The definition of rational numbers as given above seems natural, and
it has the advantage of enabling us to construct rational numbers in
terms of integers and set-theoretical concepts.
4. Note on Cantor's theory of real numbers
Let X be the set of all sequences with rational terms satisfying
Cauchy's condition of convergence. Thus, a sequence (an)ne^
of rational
terms is in X if and only if the following condition is satisfied :
(1)
for every rational number £ > 0 there is a natural number n0
such that for every natural number η and for every natural number
k the condition η > n0 implies \an — an+k\ < ε.
Let « be a binary relation on X defined thus :
(2)
(<tn)ne.r
~
0 „ W "
if and only if \im(an-bn)
n = co
= 0.
It can be proved that the relation » on X so defined is an equivalence
relation. In the case now under consideration the method of identification of equivalent elements is Cantor's method of constructing real
numbers. Real numbers are then defined as equivalence classes of the
relation « on X, i.e., classes 11(a n) n e^ 11, whose elements are sequences
EXERCISES
97
a
( n)/IE. r with rational terms and satisfying condition (1). A class
determines a real number to which all sequences (bn)ne^
terms and satisfying condition (1) and such that (a„)neJtr
verge.
||(α„), Ι 6^·||
with rational
~ (b„)ne.A~, con-
Exercises
X
1. Let X = { 1 , 2 , 3 , 4 } and let o be a relation on the family 2 of all subsets of
X defined thus: Α ρ BoN(A)
= N(B), where N(C) denotes the number of elements
of the set C for every C e l . Prove that that relation is an equivalence relation.
Indicate the equivalence class of that relation with the representative { 1 , 2 } .
2. Prove that in the set {xe
0 < χ < 2} there is no equivalence relation
having as its equivalence classes the sets: {x e 0t\ 0 < χ < 1}, {xe0t\
{xeffl:
1 <x<
—
1 < Λ:< 2}.
3. Give an example of a set with an equivalence relation defined on it and such
that each of its equivalence classes has a different number of elements and that,
for every natural number //, there is an equivalence class with exactly η elements.
Hint. Use Exercise 1.
4. Define on the set X = {x G 0ί : 0 < χ < 2} an equivalence relation such that
its equivalence classes are the sets {xe0t\ 0 < JC < 1}, {x}, {xe 0t\ 1 < Λ: < 2 } .
5. Prove that if Χ φ Ο and if (At)tGT
that satisfies the following conditions:
(a) At φ Ο for every
(b) for every pair tl912
is any non-empty family of subsets of X
teT,
of elements of T, if Atl φ At2, then Ati η At2 = O,
(c) U AT = X,
teT
then a relation ο on X which is defined thus :
xQyo\/
teT
(*e AtAye
At)
is an equivalence relation whose equivalence classes are the sets At, te T.
6. Let
be the set of all those complex numbers whose real part is other than 0.
Prove that a relation o defined on ^ * thus:
(a+bi) ρ (c+di)oac
> 0
is an equivalence relation. Give the geometrical interpretation of the equivalence
classes of that relation.
7. What is the geometrical interpretation of the equivalence classes of an equivalence relation ä defined on the set # of all complex numbers thus:
Zi x z2oArgz1 = Argz 2?
98
VII.
EQUIVALENCE RELATIONS
8. On the set of three-term sequences whose terms may equal 0 or 1 a relation ο is defined as follows:
(αΐ9α2,α3)ρ
(bit b2, b^oai
= bi
for an odd number of subscripts i = 1 , 2 , 3 . Prove that Q is an equivalence relation, and determine its equivalence classes.
9. Let Χ φ Ο be a fixed set, and let a e X be a fixed element of X. A relation »
X
on the family 2 is defined thus:
A Ä Bo
A =
ΒναφΑνΒ.
Prove that Ä is an equivalence relation and determine its equivalence classes.
2
10. Let LQ be a fixed straight line on the Euclidean plane $ . A relation Ä on the
2
set
of all straight lines on the plane 0l is defined thus:
Li ^ LZOL^LQ
Φ OAL2nL0
Determine whether χ is an equivalence relation on J£f.
Φ Ο.
CHAPTER VII
POWERS OF SETS
1. Equipotent sets. Power of a set
The concept of equipotent sets, i.e., sets of equal power, will be
introduced in the present chapter. T h a t concept is one of the most important notions in set theory
Sets X and Y are called equipotent (or equinumerous) if there is a oneto-one function / : X -> Y that maps X onto Y. The function / is said
t o establish the equipotence (or the equinumerosity) of the sets X and Y.
If X and Y are equipotent sets, then we write X ^ Y.
Examples
If X is a finite set of η elements, then a set Y is equipotent with X if
and only if Y also has η elements. The concept of equipotence is thus a
generalization — covering any sets — of the concept of the equal number
of elements in the case of finite sets. Let X be the set of all natural
numbers, and let Y be the set of all even natural numbers. The function
f: X -> Y defined by the formula f(n) = In for any natural number η is
a one-to-one function and maps X onto Y; thus it establishes the equipotence of those sets. This example shows that in the case of infinite
sets a set may be equipotent with a proper subset, i.e., with a subset
that is not identical with the set in question.
1.1. The following formulas hold for any sets Χ, Y, Z:
x~x,
(1)
(2)
X~Y=>
Y~X,
*) This concept was first systematically investigated by G. Cantor (1878), but
it had been known earlier to B. Bolzano (1851).
100
(3)
VIII. POWERS OF SETS
(X~Y)A(Y~Z)^
x~z.
Formula (1) holds, as the identity transformation Ix (cf. formula (11)
in Section 2, Chapter III) is a one-to-one function and maps X onto X,
so that it establishes the equipotence of X and X.
T o prove (2) assume that X ~ Y. Thus there is a one-to-one function / : X -> y that maps X onto Y. It follows from 2.3 and 2.2 in Chapter
l
III that the inverse function f~ : Y -> X is a one-to-one function that
maps Y onto X, which thus establishes the equipotence of Y and X.
Hence 7 ~ X.
T o prove (3) assume that X ~ Y and 7 ^ Z . Thus there are two
one-to-one functions: f: X -* Y mapping X onto Y, and g: Υ -> Ζ mapping F onto Ζ . We infer from 3.3, Chapter III, that gof:X-+Zisa,
oneto-one function that maps X onto Z, which thus establishes the equipotence of X and Z. Hence X ~ Z.
Theorem 1.1 permits the classification of sets by reference to the concept
of equipotence and thus t o generalize the concept of the number of
elements in a set so that it covers infinite sets. A cardinal number or a
power, denoted by X, is associated with every set X in such a way that
the same cardinal number is associated with two sets, X and Y, if and
only if Z a n d F a r e e q u i p o t e n t H e n c e the following formula holds:
(4)
(X = Y)o(X~
Y).
If Ζ is a finite set of η elements, then the number η is taken to be its
2
power; note also that Ο = 0 ) .
*) Cardinal numbers are introduced into set theory by means of the axiom
on the existence of cardinal numbers (see K. Kuratowski and A. Mostowski, Set
theory, Amsterdam-Warszawa 1967).
2
) Note that the theory of powers for finite sets does not contribute anything
that transcends the arithmetic of natural numbers. In such an interpretation of the
theory of finite sets the arithmetic of natural numbers is added to set theory, as
arithmetical concepts occur both in the definitions and in the theorems of set theory.
But it is possible to introduce arithmetical concepts solely by reference to set-theoretical concepts, if the system of axioms is extended correspondingly (see the book
by K. Kuratowski and A. Mostowski, referred to above).
2. ENUMERABLE SETS
101
Thus, instead of saying that sets X and Y are equipotent we may also
say that they are of equal power, or that they have the same cardinal
number.
The introduction of cardinal numbers, i.e., powers of sets, is not
indispensable. Theorems of set theory can be formulated so that n o
reference is made in them to the properties of cardinal numbers; reference is then made to relationships between cardinal numbers, and
these can be described by reference to the equipotence of sets. But the
introduction of cardinal numbers makes it possible to formulate many
theorems of set theory in a much clearer manner.
2. Enumerable sets *)
The term enumerable sets covers finite sets and those sets which are
2
equipotent with the set.yT of all natural numbers ) .
Of course, any two infinite enumerable sets are of the same power
(cf. 1.1). The power of the infinite enumerable sets is denoted by the
symbol K 0 Examples
The set of all even natural numbers is enumerable, since the function
f(n) = 2n establishes the equipotence of the set Jf of all natural numbers
and the set in question. The set of all odd natural numbers is enumerable, since the function f(n) = In— 1 establishes the equipotence of the
set^/T and the set in question.
2.1. A set Α Φ Ο is enumerable if and only if it is the set of terms of
an infinite sequence, that is, if and only if there is a function f that maps
the setJf of all natural numbers onto the set A.
am}, then it is the set of terms of an inIf A is a finite set {al9
finite sequence (bn)ne^
defined t h u s : bn = an for η = 1, ..., m , and bn
1
) The concept of enumerability and all theorems concerned with this concept
given in this section, are due to G. Cantor (1878).
2
) This definition makes sense only if the arithmetic of natural numbers is added
to set theory. Some authors use the term "enumerable'' only in connection with
sets equipotent to the set of natural numbers.
102
VIII. POWERS OF SETS
= am for η > m. If A is an infinite enumerable set, then it is equipotent
with Jf. Under the definition of equipotence there is a function / : Jf
-> A which maps^yT onto A, so that A is the set of terms of the sequence
(bn)„€jr such that bn = f(n) for « eJf. Assume now that A is the set of
terms of a sequence (bn)nçjr.
If A is a finite set, then it is enumerable
by definition. If A is an infinite set, then by taking that f(l) = bx and
f(k) is equal t o the first term in the sequence (bn)ne^r which is other
than f(k — i) for 1 - i < k, we define a function / : Jf -» A which is
a one-to-one function that maps^/T onto A. Thus A is equipotent with
Jf, and hence A is an enumerable set.
2.2. A subset of an enumerable set is an enumerable set.
Assume that A is an enumerable set and that B cz A. If Β is a finite
set, then it is enumerable by definition. If Β = A, then obviously Β is
enumerable. Assume then that Β is an infinite set, Β Φ A, and Β cz A.
Since Λ is enumerable, there is is a one-to-one function f\Jf-*A
that
maps ./Γ onto A. Let g : .yT
Β be a function defined thus : g ( l ) = /(A^),
where &χ is the least natural number such that f(kx) eB, g(n) = f(kn)
for η > 1, where kn is the least natural number such that kn > kn_x
and f(kn) e B. The function g is a one-to-one function that maps Jf
onto B. Hence Β is equipotent with Jf, and is thus an infinite enumerable set.
The following theorem refers to the union of two enumerable sets.
2.3. The union of two enumerable sets is an enumerable set.
The case in which one of the enumerable sets in question is empty
is obvious. Assume accordingly that both sets are non-empty. Let / :
Jf -+ A m a p Jf onto A and let g: Jf
Β m a p Jf onto B. Consider
a function h: Jf
A\JB defined t h u s : h(2n-\)
= f(n) and h(2n)
= g(n) for neJf.
The function h maps Jf onto AuB, and hence
by 2.1 the set A uB is enumerable.
Theorem 2.3 yields by induction that
2.4. The union of any finite number of enumerable sets is an enumerable set.
2.5. The set Jf of all integers is an enumerable set.
It follows from Theorem 2.3 that the set of all non-negative integers, being the union of the sets Jf u { 0 } , is enumerable. The set of
2. ENUMERABLE SETS
103
all negative integers is enumerable, t o o , as the function / defined by
the formula f(n) = -n is a one-to-one function that maps Jf onto
the set of all negative integers. By applying Theorem 2.3 t o t h e set
of non-negative integers and the set of negative integers we infer that 2£
is an enumerable set.
2.6. The Cartesian product of two enumerable sets is an enumerable
set.
If one of the sets is empty, then the product is the empty set. Hence
we may assume that both sets are non-empty. Let / : Jf -± A be a function that maps Jf onto a n enumerable set Α Φ Ο, and let g: Jf -> Β
be a function that maps Jf onto a n enumerable set Β Φ Ο, These
functions exist by 2.1. All elements of the product Αχ Β occur in t h e
ordered pairs (f(m), g(n)), where m, η range over all natural numbers.
r
r
Let the set of the ordered pairs (f(m), g(n)), where m eJ a n d η eJ 9
be arranged in the following infinite array:
(/Π), *(!)),
(f(D,g(2)),
(/(l),g(3)),...,
(f(2),
g(l)),
(f(2), g(2j),
(f(2), g(3)), . . . , (f(2), g(n)),
(7(3),gO)),
(/(3),if(2)),
(/(3),g(3)),...,
(f(m),
(f(m),g(2)),
(f(m),g(3)),
g(\)),
(f(l),g(n)),...
...
(f(3),g(n)),...
..., ( / ( m ) , g(n)), ...
The so-called diagonal method will be used t o define a mapping h
r
of the set Jf onto the set of all pairs ( / ( m ) , g(n)) (m eJf, η eJ ).
The ordered pairs (f(m), g(n)) such that m + n = k comprise the (k — 1 )-th
diagonal of the array given above. T h e diagonal method amounts t o
associating the successive pairs (f{m), g(«)) of the first, second, third,
..., fc-th, ... diagonal with successive natural numbers so that within
each group the pairs ( / ( m ) , g(n)) are associated in the order in which m
increases. We thus obtain
h(\) = ( / ( l ) , g(\))
h(2) =
(first diagonal),
(f(l),g(2))\
M3) = ( / ( 2 ) , g ( l ) ) i
(
S nG
C dO d
i
a
g
0
n
'
a
l
)
V N I . POWERS OF SETS
104
A(4)
(/(D^(3))
KS)
( / ( 2 ) , g(2))
A(6)
(/(3),g(l))l
A(7)
(/U),*(4)),
(third diagonal),
Every pair (f(m),g(n))
is the image of a natural number under the
mapping h. Hence /* maps the s e t ^ T onto the set AxB.
This and 2.1
imply that Α χ Β is an enumerable set.
Theorem 2.6 will be used to prove the following theorem:
2.7. The set Q of all rational numbers is enumerable.
By Theorems 2.5 and 3.6, the Cartesian product 3£ xjf, where 3£
is the set of all integers and Jf is the set of all natural numbers, is an
enumerable set. Hence there is a m a p p i n g / of Jf onto 2fχ Jf. Every
rational number can be represented as m In, where m e 3£ and
neJf.
Assume that g({m, n)) = m/n for every element (m, n) of the product
2f xJf. The function g thus defined maps 2f xjf
onto O . It follows
from Theorem 3.3 of Chapter III that g of maps Jf onto Q. F r o m
this and from Theorem 2.1 we infer that Q is an enumerable set.
2.8. Jf sets Al9
An are enumerable, then the product Aix
also is an enumerable set.
... χ An
If at least one of the sets Al9 ..-,An
is empty, then the product
Αγχ
... xAn is the empty set, too. Assume accordingly that none of
the sets Al9
A„ is empty.
The generalized product Ax χ A2 (cf. Chapter VI, Section 1) is equipotent with the product of those sets, in the sense of the definition adopted
in Chapter V, Section 1. This is so because the function / d e f i n e d by:
/ ( ( f l i , a2))
= (a9 b),
where a = ax and b — a2,
for every two-term sequence {a1, a2) such that ax e Al9 a2 e A2, establishes the equipotence of those sets. From this and 2.6 we infer that the
generalized product A1 χ A2 is an enumerable set. Assume that the
product A1x
... xAm,
where 1 < m < η, is an enumerable set, and
form the product Axx
... χAmxAm+l.
This is equipotent with the
product ( ^ x ... xAm)xAm+1.
For let g: ( ^ x ...
xAm)xAm+i
2. ENUMERABLE SETS
->^χ
... xAmxAm+1
105
be a function defined by the following
for-
mula:
where
and
for every
and every
Now. £ is a one-to-one function and maps
onto
Thus it establishes the equipotence of these
is enumerable by assumption and
sets. Since A* χ
is
enumerable, too, hence, by 2.6, the se
is enumerable. Consequently
is also enumerable. We infer
by induction that
is an enumerable set.
The following theorem generalizes 2.4 to any enumerable indexed
family of enumerable sets.
2.9. For every enumerable indexed family
r
is an enumerable set for every η eJ ,
is an enumerable
of sets
such that
the generalized
unio\
set.
It follows from 2.1 that, for every η ejV> there is a function fn;
r
Jf -> An that maps the setJ
of all natural numbers onto the set An.
Hence the images fn(m) of all natural numbers m, for all functions
r
fn, η eJ ,
cover all the elements of the sel
= f„(m) for every ordered pair (n,m)
By setting f((n,
m))
of natural numbers we define
a function / that maps the Cartesian product Jf xjf
onto the set
Since Jf xjf is an enumerable set (see 2.6), hence, by 2.1, there is
a mapping g of Jf onXoJf xjf. It follows from Theorem 3.3, Chapter
III, that the function fog
maps Jf ont<
which in view of 2.1
completes the proof of Theorem 2.9.
Theorems 2.8 and 2.9 will now be used in the proof of the following
theorem.
2.10. The set of all finite sequences with terms from a fixed enumerable set is an enumerable set.
That set is the union
An of the sets An, η eJf',
where An is the
106
VIII. POWERS OF SETS
set of all Ai-term sequences with terms in a given enumerable set A.
n
(see Chapter III, Section 1).
By definition, An = A for each neJf
n
It follows from Theorem 2.8 that the sets A , for each n G / , are enumerable. Hence the sets An, for every η eJf', are enumerable. This and 2.9
00
imply that the set \J An is enumerable, too, which completes the proof
«=1
of Theorem 2.10.
The following theorem follows from 2.10 and 2.7.
2.11. The set of all polynomials in one variable with rational coefficients
is enumerable.
n
Every polynomial α0 + αγχ + ... +anx
is determined by a finite
sequence (a0, αγ, ..., an) of rational numbers. The set of all rational
numbers is enumerable by 2.7. This and 2.10 imply that the set of all
finite sequences with rational terms is enumerable. Hence, the set of
all polynomials in one variable with rational coefficients is also enumerable.
2.12. The set of all algebraic numbers is enumerable.
Algebraic numbers are zeros of polynomials with rational coefficients. It follows from 2.11 and 2.1 that there is a m a p p i n g / of the
set Jf of all natural numbers onto the set of all polynomials in one
variable with rational coefficients. The images f(n) of all natural numbers
r
η eJ under the m a p p i n g / t h u s cover all polynomials in one variable
with rational coefficients. Let An denote the set of all the zeros of a polynomial f(n). As is known, An is a finite set for every n. The set A of
all algebraic numbers is the union of all the sets An, so that A = {J An.
neJf
From this and from 2.9 we infer that the set A is enumerable, which
completes the proof of Theorem 2.12.
3. Examples of non-enumerable sets
A set which is not enumerable is called non-enumerable. Let ^
denote the set of all real numbers. For any real numbers a, b such that
a < b,
b} will be used to denote the set {x e 0t\ a <; χ < Z?}, i.e.
the closed interval which starts with a and ends with b.
We also prove
3. EXAMPLES OF NON-ENUMERABLE SETS
107
3.1. The set of the real numbers in the interval <0, 1> is non-enumerx
able ).
T o prove 3.1 it will be shown that there is n o sequence of terms
from the interval <0, 1> which contains every real number in <0, 1>.
Let (cn)nejr
be any sequence that satisfies the condition
0 < cn < 1
(1)
for every η eJf.
Let the interval <0, 1> be denoted by {a0,b0}.
F r o m among the
intervals <0, y > , < y , -§->, <-§-, 1> we select one which does not contain
cl and we denote it by <TFI,i>I>. T h e following conditions are then
satisfied:
(2)
c^ia^bà,
= |,
bi-a1
c <0, 1> = < a 0 , b0}.
(fl^b^
By dealing similarly with the interval <α χ , Z^) we obtain an interval
< ö 2 , b2y satisfying the conditions:
(3)
c2<ß(a2,b2y,
b2-a2
2
< a 2 , b2} cz < a l 5 b^.
= \/3 ,
In general, if we have a given interval < α „ _ Ι , £ „ - I > >
Cn-l
(4)
A - l > ,
=
n
ί
^
2 , such that
β"~\
<fl„_I, 6 „ - I > CZ <tf„_ 2, 6„_ 2 >,
then, by continuing the same procedure, we determine an interval
<a„, b„y that satisfies the conditions:
(5)
cn<£(an9bn},
bn-an
= 1/3",
<an,b„)
cz (a^,
bn_t}.
In this way we define a sequence of intervals
bn})ne^
satisfying conditions (5) for every η > 1. It follows from (5) that the following inequalities hold for every η eJf\
(6)
0 < an < an+l
< bn+i
^bn
< 1.
The sequences (an)„e^ a n d (bn)n(=jr
are thus monotonie a n d bounded,
and hence are convergent. By (5), bn — a„ < 1/3" for every
neJf.
Hence lim(6„ — an) = 0 . Consequently, lima„ = \imbn = c , where c
#I"*OO
η = oo
n = oo
is a real number from the interval <0, 1>. The number c is in every
l
) This theorem is due to G. Cantor in 1874.
108
VIII. POWERS OF SETS
interval (a„,bn},
and hence it differs from every cn, since by (5) cn
φ (a„, bn} for every η eJf. Since the sequence (cn)ne^
has been selected
in an arbitrary manner, we have thus proved that for every sequence
with terms from the interval <0, 1> there is a real number in this interval which is not a term of that sequence, which completes the proof
of Theorem 3.1.
3.2. If a set A is non-enumerable and if A cz B, then Β is also nonenumerable.
Theorem 3.2 follows directly from the definition of non-enumerable
sets and from 2.2.
It follows from Theorems 3.1 and 3.2 that
3.3. The set 0t of all real numbers is
We now show that
non-enumerable.
3.4. The set of all irrational numbers is non-enumerable.
Should that set be enumerable, then the set of all real numbers,
as the union of the set of rational numbers and the set of irrational
numbers would, under Theorems 2.7 and 2.3, be enumerable, which
contradicts Theorem 3.3.
Likewise, we show that
3.5. The set of all transcendental numbers is
non-enumerable.
Transcendental numbers are those real numbers which are not
algebraic. The set of all algebraic numbers is enumerable (see 2.12).
Should the set of all transcendental numbers be enumerable, then the
set of all real numbers as the union of two enumerable sets would, by
Theorem 2.3, be enumerable. This, however, contradicts Theorem 3.3.
Theorem 3.5 states, in particular, that there are transcendental
numbers. Examples of transcendental numbers are found by a quite
different procedure; the same holds for the proofs of transcendence of
certain well-known numbers, such as e and π .
4. Inequalities for cardinal numbers. The Cantor-Bernstein theorem
Let A = η and Β = m. We say that the cardinal number η is not
greater than the cardinal number m if the set A is equipotent with
4. INEQUALITIES FOR CARDINAL NUMBERS
109
a subset of the set B. Then every set of power π is equipotent with
a subset of a set of power m. This is written:
(1)
n<m.
If η < m and η φ m, then we say that the cardinal number η is less
than the cardinal number m and write:
(2)
η < m.
Example
The set Jf of all natural numbers is equipotent with a subset of
the set 0t of all real numbers, namely with the %t\Jf <= 0t. Theorem
3.3 states that the set 01 is non-enumerable, and hence its power is
different from that of the set Jf. Thus
(3)
Jf < m.
The power Jf is denoted by the symbol K 0 (cf. Section 2). The power
of the set 0t of all real numbers is called the power of the continuum
and denoted by c. Thus it follows from (3) that
(4)
Ko <
c
4.1. For any cardinal numbers n, m, ρ the following formulas
(5)
η < n,
(6)
if η < m and m < ρ, then η < ρ.
hold:
Let A be any set of the power n. By 1.1 (1), A ~ A cz A. This
formula yields that Ä^Ä~,
i.e., η < n. Assume now that A = n,
Β = m, C = p, and that η < m and m < p. It follows that A is equipotent with a subset B^ of B, and that Β is equipotent with a subset
Q of C. Thus, there is a one-to-one function / : A -> Bt that maps A
onto 2?!, and a one-to-one function g : Β
C x that maps 5 onto Q .
Let #12?! be the function g restricted to B1 (see Chapter III, Section 1, (5)).
The function g\Bt also is one-to-one and it maps B1 into Ct. By Theorem
3.3 in Chapter III we infer that the composition (g|2?i) of is a one-toone function that maps A into Cx <= C. Let C 2 <= d be the image
of A under this function, which then maps A onto C 2 . Hence
VIII. POWERS OF SETS
110
we infer that A is equipotent with C2 <= C, which proves that η < p.
We now prove the following Cantor-Bernstein
1
theorem ):
4.2. For any cardinal numbers xx, m:
(7)
/ / η < m and m < η , then η = m.
Let A be any set of power n, so that A = n. Since m < η , A contains
a subset Β of power m. Thus we have
(8)
Β cz A
and
Β = m.
We also have η < m, so that A is equipotent with a subset of B. Thus,
there is a one-to-one function / : A -> B. It follows from this fact and
from (8) that
(9)
f(A)
cz Β cz A.
T o prove that η = m it suffices to define a one-to-one function
g: A -> Β that maps A onto B. T o do so we first resort to certain auxiliary constructions.
Let
(10)
C =
B-f(A).
It follows from (10) and 4.1 (3) in Chapter I that C czB. This and (9)
yield
(11)
CczA.
By applying to (11) Theorem 5.9, Chapter VI, and (9) we infer that
f(C) czf(A) cz A. Hence
(12)
ÄQczA.
We now define by induction a sequence {Cn)nsjr
(13)
d
of subsets of A :
=f(C)czA.
Assume that we have defined Cn cz A. We then define
(14)
J
C „ +1 =
f{Cn\
) Also called the Schröder-Bernstein theorem. The first correct proof of this
theorem is due to Bernstein and was published in Leçons sur la théorie des fonctions
by E. Borel, Paris 1898.
4. INEQUALITIES FOR CARDINAL NUMBERS
1\ \
Since by assumption Cn cz A, hence by 5.9, Chapter VI, and (9) we
infer that f(Cn) czf(A) c A. Hence C „ + 1 cz A.
It follows from 2.1 (2), Chapter IV, that ( J Cn cz A.
This and
n= l
(11) yield
C u U Q
ŒA.
n= l
Let
i) = C u U C „ .
(15)
By the previous inclusion we have
(16)
DczA.
We can now form the image of the set D under the m a p p i n g / . By applying (15), 5.2 in Chapter VI, and (13), (14), we obtain
f(D)
= / ( C ) u Û / ( C „ ) = Cx u Û Cn = u
n=l
n=z2
η
C„.
-1
This and (15) >ield
(17)
D =
Cvf(D).
It follows from (16) and 4.3 (10), Chapter I, that
(18)
A =
Du(A-D).
The function g will now be defined t h u s :
Î
a
for a G D,
f(a)
iorasA-D.
We first show that g maps A onto B, i.e., that
(20)
g(A) = B.
It follows from the definition of g that
(21)
g(D) = D,
g(A-D)=f(A-D).
By (18), and 5.1 in Chapter VI, and by applying (21) we obtain
(22)
g(A) = g(D KJ(A-
D)) = g(D) vg(A -D)
= D uf(A - D).
1J 2
VIII. POWERS OF SETS
It follows from (22), (17), 5.1 in Chapter VI, (18) and (10) that
(23)
g(A) = C u / ( D ) uf(A - D) = C u/(Z> u (A - D))
= Cvf(A)
=
(B-f(A))vf(A).
Since, by (9), f(A) c B, hence it follows from 4.3 (10) in Chapter I,
t h a t / 0 4 ) u(B-f{A))
= B. This and (23) yield g(A) = B, which proves
(20).
We now show that
(24)
As fis
that
g(D)ng(A-D)
= O.
a one-to-one function we infer, using (21) and 5.8, Chapter VI,
(25)
g(A -D)
= f(A -D)
=
f(A)-f(D).
At the same time from (10) we obtain f(A) nC = f(A) n(B-f(A))
Hence it follows that
(26)
= O.
f(A)=f(A)-C.
Now (25), (26), 4.5 in Chapter I and (17) yield
(27)
g(A -D)
= (f(A) - C) -AD)
= f(A) - ( C uf(D))
= f(A)-
D.
Formulas (21) and (27) yield
(28)
g(D) ng(A -D)
= D n(f(A)
-D)
= O,
which proves (24).
Formula (24) will now be used to prove that g is a one-to-one function. Assume that ax e A, a2 e A, and αγφ a2. We show that
(29)
g(a1)^g(a2)-
lfa1 and a2 both are in Z>, then (29) is satisfied, since by (19) g(ax) =
and g(a2) = a2. If a± and a2 both are in Α-D,
then (29) is satisfied,
since by (19) g(ax) = f(ax) and g(a2) = f(a2),
and / is a one-to-one
function. And if at e D and a2 eA-D,
then (29) is satisfied in view
of (24).
The proof of Theorem 4.2 is thus complete.
The Cantor-Bernstein theorem is also often formulated in the following way:
113
5. SETS OF THE POWER OF THE CONTINUUM
4.3. For any sets A, B, C:
(30)
if A cz Β cz C and Ä = C, then A = Έ = C.
If A cz Β cz C, then Z < Β
we have Ä ·•[ Β and Β <A.
: C . If at the same time Ä = C, then
It follows from 4.2 that A = E9 which
proves 4.3.
Conversely, 4.2 follows easily from 4.3. F o r let η < m and m < η ,
and let C be any set of power n. Since m < η , C contains a subset Β
of power m. Thus Β cz C and Β = m. Since at the same time η < m,
Β contains a subset A of power n. Thus we have A cz Β cz C ,
Ä = C = n. It follows from 4.3 that Β = A, so that m = n.
5. Sets of the power of the continuum
We said in Section 4 that the power (cardinal number) of the set
0t of all real numbers is called the power of the continuum and denoted
by c. We have also shown that K 0 < c. Thus there are at least two
different infinite cardinal numbers: the power of the set of all natural
numbers and the power of the set of all real numbers.
Other examples of sets of the power of the continuum will be given
below.
5.1. The open interval { j c e l : — ·~π < χ < \π)
set M of the power of the continuum.
is a subset of tht
A function / : {x e 0ί\ —\iz < χ < } π } -*
defined by the
formula: f(x) = t a n x for every χ in that interval, is a one-to-one function; it maps this interval onto 0t, and hence establishes the equipotence
of these two sets, which proves Theorem 5.1.
5.2. Every open interval {x G 0l\ a < χ < b},
the power of the continuum.
where a < b, is of
1
The function / defined by the formula f(x) = (b — α ) π ~ ( * + | π ) + a,
for those x's which satisfy the condition — \π < χ < \π, is a oneto-one function that maps the open interval {x e 0t\ — γ π < χ < -2-π}
onto the open interval {x e 01 : a < χ < b}. Thus it establishes the
equipotence of those intervals. This fact and 5.1 yield 5.2.
U4
νίΓΓ. POWERS OF SETS
5.3. Every closed interval {x e 0ί\
of the power of the continuum *).
a Sχ
b},
where a < b,
is
T o prove this theorem note that
{x e 0i\ a < χ < b) cz {x e &\ a ; χ
Γ b} cz
0t.
Theorem 5.3 follows directly from the above formula and from Theorems
5.2 and 4.3.
The following theorem makes it possible to obtain further examples
of sets of the power of the continuum.
5.4. If A is a set of the power of the continuum and if Β is an enumerable subset of A, then the set difference A —Β is a set of the power of
the continuum.
Assume that Β = K 0 ·
L e t / : Β -+ (A-Β) be a one-to-one mapping of Β into A — B. Such
2
a mapping does exist, as A—B is a non-enumerable subset of
A ).
Hence
(1)
f(B) cz Α-B,
f(B) = X 0 ,
Bnf(B)
= O.
The set Buf(B) is an infinite enumerable subset of A. Hence there is
a one-to-one function g: Β <uf(B) -> f(B) that maps Buf(B)
onto
f{B). The set A can be presented as the union of two disjoint sets (Chapter I, 4.3 (10))
(2)
A=(A-(B
uf(B)))
u(2?
uf(B)).
In view of (2) we can define a mapping h : A
χ
( 3)
)h ( x
\g(x)
for χ
for
A thus :
eA-(Bvf(B)),
xeBuf(B).
The mapping h is one-to-one. To prove this assume that χγ φ x2.
If x1 and x2 are both in A — (Bvf(Bf),
then A ( * i ) = * i and h(x2) = x2,
!
) One may show similarly that every interval which is closed at one end is of
the power of the continuum.
2
) The set A—B, being non-enumerable, is infinite, and every infinite set is
of a power greater than, or equal to, X 0 (see Chapter XI, Theorem 7.4). Hence A — B
has a subset of power X 0 , which proves the existence of a one-to-one mapping
/ of the set Β into A-B.
115
5. SETS OF THE POWER OF THE CONTINUUM
and hence h(xx) φ h{x2). If * i and x2 are both in Èvf(B),
then A f o )
a n (
=
* Ä f e ) = # ( * 2 ) > and g(xj) φ g(x2) since g is a one-to-one
If,
finally,
e A — (2?u/(2?)) and
mapping. Hence Η{χ1)φΗ{χ2).
x2 Ε Β u / ( Ä ) , then A f o ) = ^ e Λ - ( 5 u / ( £ ) ) and A(JC 2 ) = g ( x 2 ) e / ( £ ) .
Since the sets A - (B vf(Bj) and f(B) are disjoint, we have h(xx) φ h(x2).
The function h maps A onto A—B, for by (2), (3), 5.1, Chapter V,
and (1) we have
h(A) = h(A - (B uf(B)))
ug(B uf(B))
uh(B u / ( 5 ) ) = (A - (B uf(B)))
= ((A — B) n(A-f{B)))
= ((A-B)
vf(B))
n((A-f(B))
= (A-B)
ηA =
A-B.
u
uf(B)
uf(B))
Thus the mapping h establishes the equipotence of the sets A and A — B.
Hence it follows that A — B is a set of the power of the continuum.
If Β is a finite subset of A, then there is an infinite enumerable subset
B1 c A such that Β cz Bx.
Then A-Bx
c z A — B cz A. Since
A-B\
= c = A, hence by applying 4.3 we obtain A — B = c.
5.5. 77ze set of all irrational numbers is of the power of the continuum.
The set of all irrational numbers is the difference St — Q of the
set M of all real numbers, which is of the power of the continuum,
and the set Q of all rational numbers, which is of power K 0 · This and
5.4 yield that the set of all irrational numbers is of the power of the
continuum.
By 5.2, 5.3, and 5.4, we likewise conclude that
5.6. The set of all irrational numbers in any (non-empty) interval
is of the power of the continuum.
By applying Theorems 2.12 and 5.4 and by reasoning in the same
way as in the proof of 5.5 we conclude that
5.7. The set of all transcendental numbers is of the power of the continuum.
The following theorem, to be referred to in the next section, will
now be proved.
VIII. POWERS OF SETS
116
5.8. The set of all functions f: Jf -* {0, 1} (that is, the set of all
sequences (an)nçjr
such that an e {0, 1} for every η eJf) is of the power
of the continuum.
Let A be the set of all sequences (an)nejr
with terms in the set {0, 1}.
The function g with real values will be defined on A as follows:
if an = 0 for infinitely many n,
n=l
g((ßn)nejr)
=
1+
Σ
n
2
if an = 0 for finitely many η or if an φ 0
for every η e Jf.
n=l
As is known g is a one-to-one function. Hence it follows that A = g(A).
It is also known that
{xe0t:
0<x < l } c g ( 4
It follows from Theorems 5.2 a n d 4.3 and from the formula above
that g(A) = c. Hence Ä = c.
The set of all sequences whose terms are natural numbers, and even
the set of all sequences whose terms are real numbers, are sets of the
power of the continuum
The search for other cardinal numbers that are powers of subsets of
the set 0i has generated the following problem: is every subset of the
set ^ of all real numbers either enumerable or of the power of the
continuum? The hypothesis stating that the answer t o this question is
in the affirmative, formulated by Cantor, is called the continuum hypothesis. We shall revert t o this issue in Chapter XI, Section 9.
6. The power set. Cantor's theorem. Consequences of Cantor's theorem
As stated previously (see Chapter III, Section 1), the set of all funcX
X
tions / : X -> Y is denoted by Y . In particular, {0, 1} denotes the set
of all f u n c t i o n s / : X
( 0 , 1}. Such functions are called characteristic
l
) Proofs of these theorems may be found in Set theory by K. Kuratowski and
A. Mostowski (see footnote on p. 100).
6. POWER SET. CANTOR'S THEOREM
117
functions of the subsets of X. Let A cz X. The f u n c t i o n / ^ : X -» {0, 1 } ,
defined t h u s :
Î
1
for χ e A,
0
for X φ A,
is the characteristic function of the subset A of the set X.
We show that
6.1. For every set X, the set of all subsets of X is equipotent with the
set {0, 1}*
Assume, for every A cz X,
(2)
g(A)=fA.
We have thus defined a function g that maps the set of all subsets of
X
X onto {0, 1}*. F o r let / b e any function in {0, 1} . Consider the set
of those elements χ oï X for which / takes the value 1, that is, the inverse image of the set {1} under the function / . Let this set be denoted
by A. We thus have
A = {xeX:f(x)
= 1}.
It can easily be seen that / is the characteristic function of A, so that
which proves that g is a mapping onto
f = fA> Hence g(A) =fA=f,
X
the set {0, 1 } . It will be shown that g is one-to-one. T o d o so assume
that A cz X9 Β cz X, and Α Φ Β. There is then an element χ such which
is in A and is not in B, or is in Β and is not in A. Assume that χ φ A
and x e B. Then fA(x) = 0 and fB(x) = 1. Hence, fA Φ fß, so that g(A)
Φ g{B). Thus the function g establishes the equipotence of the family
x
of all subsets of A" with the set ( 0 , \ } .
Theorem 6.1 suggests that the set of all subsets of X be denoted by
X
1
2 . The set 2 is called the power set of the set A" ).
It was proved in the preceding section (Theorem 5.8) that the set
{0,1}*^ is of the power of the continuum. This a n d Theorem 6.1 yield
immediately that the set of all subsets of the set Jf of all natural numbers
is of the power of the continuum. We have thus proved that
X
6.2. The set of all subsets of the setJf
(3)
W
l
) See p. 23.
= c.
is of power c, so that
VIII. POWERS OF SET F
118
It follows from Theorem 6.2 that
6.3. Iff
= K 0 , thenW
= c.
Assume that X = K 0 . Hence there is a one-to-one function / that
maps Jf onto X. By setting, for each A c / , #04) = / ( Λ ) we have
r
This follows from the fact
defined a function g that maps 2 - onto
that / maps Jf onto X, hence every subset of X is the image of a set
A czjf under / . The function g is one-to-one, for assume that A cz Jf ^
Β czjf,
and Α φ B\ there is then an η Ejf such that « is exactly in
one of the sets A, B. Suppose, foi instance, that n e A and η φ Β. Hence
it follows t h a t / ( w ) ef(A) a n d / ( / i )
For should / ( « ) ef(B),
there
would be a natural number m e Β such that f(n) = f(m). This, as / is
a one-to-one function, would yield η = m, which is impossible as η φ Β
and m Ε Β. We have thus proved that f(A) φ f(B), from which it follows
r
that g(A) Φ g(B). Thus g establishes the equipotence of the sets 2- and
X
2 . This and 6.2 yield 6.3.
Theorem 6.3 and formula (4) in Section 4 imply
6.4. For every set X of power K 0
Χ <
(4)
X
2.
At the same time it is known that if X is a finite set of η elements,
then 2
X
n
is a set of 2 elements (Chapter II, Theorem 2.1). Here again
X
X < 2 . This inequality also holds if X is the empty set, for the power
of the empty set equals 0 , and the power of the set of all subsets of the
empty set equals 1. Cantor's theorem generalizes this equality so that
it holds for all sets.
6.5. CANTOR'S THEOREM. For every set
Χ <
(5)
X
X
2.
If X = Ο this theorem is, as has been shown above, true. Assume
X
now that Χ Φ Ο. It can easily be demonstrated that X < 2 .
X
g(x) = {x} for every χ Ε X. The function g: Χ -> 2
Assume
is one-to-one and
maps X onto the family of all one-element subsets of X, that family being
X
contained in 2 . This proves the inequality X
that Î Φ Ψ.
It remains to prove
119
6. POWER SET. CANTOR'S THEOREM
Assume now that, for a set Χ Φ 0 , there is a non-empty subset
X
A α X such that A ~ 2 . Hence it follows that there is a one-to-one
X
f u n c t i o n / : A -> 2 that maps A onto 2*. For every x e A,f(x) is a subX
set of
and hence an element of 2 . Hence two cases are possible:
either χ e f(x),
or χ φ f(x).
Let Ζ = { x e ^ : i ^ / ( * ) } . Obviously,
Ζ cz X. Moreover, it follows from the definition of Ζ that, for every
χ e A,
(x e Ζ ) ο
(6)
(*£/(*)).
As / maps ,4 onto 2*,- every subset of X is the value of the function /
for an x e A ; in particular, there is an a e A such that Ζ = f(a). Consider now whether a e Ζ or α φ Ζ . It follows from (6) that if a e Z ,
then α £ f(a) = Z . If, on the contrary, α £ Ζ , then a e f(a) = Z. This
results in a contradiction. We have thus proved that no non-empty
X
subset of a set X is equipotent with the set 2 \ in particular, X is not
X
equipotent with 2 . This completes the proof of (5).
Cantor's theorem makes it possible to construct sets of increasingly
high powers. For instance, taking the set Jf of all natural numbers as
r
the starting point we construct the s e t s : J ,
Cantor's theorem and formula (3) yield that
K 0 =Jr
<
= c < W
< 2
lJ/
2·^, 2 \
2 l Jr
2 2
2 ^ ,
< ...
In this way we obtain infinitely many different cardinal numbers.
It also follows from Cantor's theorem that there is no set of all
1
Z
sets ) . For should there be a set Ζ of all sets, then the family 2 of all
Z
subsets of Ζ would have to be itself a subset of Z . Hence the set 2
would have to be equipotent with a subset of Z , which is not possible,
since we have demonstrated in the proof of Cantor's theorem that, for
every set X, the family of all subsets of X is not equipotent with any
subset of X.
x
) On adopting a system of axioms for set theory which includes the concept
of class (see Chapter I, Section 10 in fine, p. 24-25) we can use such concepts as
the class of all sets, the class of all cardinal numbers, etc. Such classes, however,
are not sets, but merely share certain properties of sets, so that no paradoxes arise^
VIII. POWERS OF SETS
120
Exercises
1. Is a set whose every proper subset is enumerable itself enumerable?
2. Determine the power of the set of all those circles in the plane whose centres
have rational coordinates and whose radii have lengths which are integer multiple
of j / 2 .
3. Determine the power of the set of all equilateral triangles in the plane whose
side equals a > 0 and which have the point (0, 0) as one of the vertices.
4. Prove that the set of all equilateral triangles in the plane whose centre of
gravity is at the origin of the system of coordinates and whose one of the vertices has
rational coordinates is enumerable.
5. Prove that the set of all intervals of the real line with rational end points is
enumerable.
6. Prove that every family of disjoint intervals of the real line is enumerable.
Hint. Refer to Exercise 5.
7. Prove that
(a) for any sets Alt Bit A2, B2,
that (AiXAi) ~
(BiXB2);
the condition Αγ ~ Bx and A2 ~ B2 implies
(b) for any sets A, B, if A ~ B, then 2
A
B
^ 2;
A B
(c) if the sets A and Β are disjoint, then, for every set X,X ^
AkjB
that if AnB=
O, then 2
~ 2^x2*.
A
B
~ X xX .
Infer
8. Prove that the Cartesian product of two sets of the power of the continuum
is itself a set of the power of the continuum.
Hint. Use Theorem 6.2 and Exercise 7.
9. Prove that if X > X 0 , then X contains a subset A which is enumerable and
such that X-A ~ X.
10. Prove that X 0 < X is a necessary and sufficient condition for the set X to
be equipotent with a proper subset of itself (i.e., a subset different from X).
CHAPTER IX
ORDERED SETS
1. Ordering relations
1
A relation ρ a XxXis
said t o be an ordering relation on the set X )
if it is reflexive, antisymmetric, and transitive (see Chapter V, Section 4),
i.e., if it satisfies the following conditions:
(r)
χ ρχ
(a)
(χργΑγρχ)=>χ
(t)
(χργΛγρζ)=>χρζ
for every x e X,
= γ
for every x,y
GX,
for every x, y,
z G Χ.
Instead of χ ρ y we often write x
which is read: χ is contained
in y or y contains x.
If ρ cz Χ χ Χ is an ordering relation on X, then we also say that ρ
2
orders X, and the ordered pair (Χ, ρ ) is called an ordered set ).
Examples
(I) Let 91 be any non-empty family of subsets of a set Χ φ Ο, and
let ρ be the inclusion relation on 91. This relation orders 91, since, for
any sets A, B, C that are in 91, we have, in view of Theorem 1.1 (9), (10),
(11), Chapter I, A cz A; if A cz Β and Β cz A, then A = B; if A c z Β
and Β cz C, then A cz C. Hence we infer that the family 91 is ordered
by the inclusion relation.
(II) The set Jf of all natural numbers is ordered by the divisibility
relation | (n\m if and only if η is a divisor of m). This is so because the
*) Relations which satisfy conditions (r), (a), (t) are called partial ordering relations by some authors.
2
) The theory of ordered set was initiated by R. Dedekind in the late 19th century.
It has played an important role in modern mathematics.
IX. ORDERED SETS
122
following formulas hold for any natural numbers n, m, k:
n\m,
(n\m Am\n) => η = m,
(n\m A m\k) => n\k.
(III) The set
of all functions that map the set ^ of all real
numbers into 01 is ordered by the relation
defined thus: / ig if
and only if, for every xe 0t, fix)
gix). The relation is reflexive,
i . e . , / / , since, for every x G 0t, fix)
fix). The relation is antisymM
g(x)
metric, since, for any functions / g in 0i , if/-... g (i.e., fix)
for every Λ: G 0t) and if g
/ (i.e., g(x)
fix) for every χ e 01), then
f{x) = gix) for every χ e 01, which means (see Chapter III, Section 1,
formula (4)) that / = g. Finally, the relation is transitive, since, for any
m
g (i.e., fix)
gix) for every x G M) and
functions / g, h in M , iff
g
h (i.e., gix)
liix) for every x G 01), then /
h (i.e., fix)
hix)
for every x e $ ) .
(IV) Every non-empty subset X of the set 01 of all real numbers is
ordered by the relation . ("not greater than"), since, for any real
numbers x, y, ζ in X, the following conditions are satisfied:
x
x,
(x
y Ay
χ) => χ = y,
(χ
ν λ y < ζ) => χ < ζ.
(V) If Χ is a finite set, then any relation
which orders X can be
described by reference to a geometrical illustration by means of diagrams.
This will be explained by a number of examples. Figures 11-15 illustrate
relations that order certain finite sets.
Fig.
11
Fig.
12
Fig.
13
Fig.
14
Fig.
15
Figure 11 illustrates a relation ^ that orders the set X =
{a,b}.
The relation is reflexive, hence a
a and b
b, and moreover, a
b,
as the points representing a and b are connected by a segment so that
1. ORDERING RELATIONS
123
the point representing a is situated below the point that represents b.
It can also easily be checked that the relation now under consideration
is antisymmetric and transitive.
Figure 12 illustrates a relation
that orders the set X = {a, b, c}.
This relation is reflexive, hence a < a, b ^ b, and c < c, and moreover,
b < a and b < c, as the pairs of points which represent b, a and b, c,
respectively, are connected by segments so that the point which represents b is situated lower than those which represent a and c. The
relation in question also satisfies conditions (a) and (t).
Figure 13 illustrates a relation < that orders the set X = {a, b, c) and
is defined thus: as in previous cases, a < a, b
b, and c < c, and
moreover, a <,b and c <b. Conditions (t) and (a) are satisfied, too.
Figure 14 illustrates a relation - that orders the set X = {a,b,
c,d}
and is defined t h u s : as in previous cases, the relation is reflexive, so
that a - a, b • b, c • c, d
d, and moreover, a
b, a ; d, b
c,
d
c, and also a • c, which follows from the previous formulas and
from the assumption that the relation in question is transitive. The antisymmetry of the relation can easily be checked.
Figure 15 illustrates a relation ; that orders the set X = {al9 a2, a3,
a 4 , tf5, a6} and is defined t h u s : as in the previous cases, the relation is
reflexive, so that a,
a} for every j = 1, 2, 3 , 4, 5, 6; moreover, a 4
. a5, a 4
a6, a3 • aA, ax • a3, and a2 · a3. These formulas and the
assumption that
is a transitive relation yield: a3
a5, a3
a6, ax
[ a 4 , a2 ; α 4 , αλ
a5, ax ·.' #6, a2 < a5, a2 : a6. Antisymmetry can
easily be checked. It is also worth noting that the formula at ; ty, for
/ Φ j , holds if and only if a broken line rises from the point representing α ι to the point representing aj.
The following theorem establishes a relationship between those relations which order a given set X and those relations which are irreflexive
and transitive on X.
1.1. If
is a relation that orders a set X, then the relation -< defined
on X in the following way.
(1)
χ < y if and only if χ - y and χ φ y for every x, y Ε Χ,
is irreflexive
asymmetric.
and transitive
and hence (see Chapter V, Theorem 4.1)
124
IX. ORDERED SETS
Conversely, if < is an irreflexive and transitive relation on X, then
the relation ^on X which is defined by:
(2)
Λ: < y if and only if χ <y or χ = y for every x,y
eX,
orders the set X.
T o prove the first part of Theorem 1.1 assume that < is a relation
that orders the set X and that -< is the relation defined by formula (1).
The formula χ < x cannot hold since, by (1), this could occur only if
χ < χ and χ φ χ. Hence the relation < is irreflexive. Assume further
that χ <y and y < ζ. This and (1) yield that χ < y, y < ζ, χ Φ y, and
y φ ζ. Since < is a transitive relation, hence it follows from these conditions that χ < z. Should χ = ζ, then the previous conditions would
yield that ζ <>> and y < ζ, which, in view of the antisymmetry of the
relation < , would imply y = ζ contrary t o the assumption that y Φ ζ.
Hence χ φ ζ. This and χ < ζ imply χ -< ζ. We have thus proved that
-< is a transitive relation, which completes the proof of the first part
of Theorem 1.1.
Assume now that -< is an irreflexive and transitive relation on X.
Let < be the relation on X defined by (2). The reflexivity of < follows
directly from its definition. Assume now that χ
and j> < z. This
and (2) yield that (x < y or χ = y) and (y -< ζ or y = z). If χ -< y and
y -< ζ, then χ -< ζ, and hence χ < ζ. If χ = y and j = ζ, then χ = ζ,
and hence Λ: < z. If χ -< j and >' = ζ, then * -< ζ, and hence χ < ζ. If
χ — y and j> -< z, then χ < ζ, and hence χ < ζ. We have thus proved
that < is a transitive relation. Suppose now that χ < j and j < x . This
and (2) yield that (x ^y or χ = y) and (y < χ or y = χ). The conditions χ -< j and j> •< x are mutually exclusive, as the relation ^ is
asymmetric (see Chapter V, Theorem 4.1). Likewise, the conditions χ <y
and y = χ are mutually exclusive, and so are the conditions χ = y and
y <x. This leaves only one possible case, namely x = y and j = x .
Thus Λ: y and j
Λ: imply x = y, which proves that < is an
asymmetric relation and completes the proof of Theorem 1.1.
F o r a given relation
that orders a set X we shall often consider
the relation •< defined by formula (1).
Given an ordered set (X, < ) it will be said that χ precedes y (where
x e X and y e X) if and only if χ ^ j ; and χ φ y, that is, if χ < y, where
-< is the relation on X defined by formula (1).
2. MAXIMAL AND MINIMAL ELEMENTS
125
Examples
In example (II) 3 precedes 6 as 3|6 and 3 φ 6. This fact is recorded
t h u s : 3 -< 6. In example (III) let / b e the function defined by the formula
2
2
f(x) = x , and let g be the function defined by the formula g(x) = x +
+ 1. Then / < g and / φ g. Hence / precedes g, which is written as:
f<g.
2. Maximal and minimal elements
Let (X, < ) be an ordered set. An element x0 e X is called maximal if
it does not precede any element in X, so that there is n o x e X such that
x0 -< x. In symbols:
~ V (*o - < * ) ·
(1)
xeX
In accordance with the adopted definition of < (see Section 1, formula
(1)), condition (1) may be written without using of the symbol -< as
(2)
-
V
xeX
(x0
< ΧΑΧ0Φ
χ).
Examples
(I) Let 91 be any non-empty family of subsets of a set Χ Φ Ο, such
that it contains the set X. The family 91 is ordered by the inclusion relation c . In the ordered set (91, cz) there is exactly one maximal element,
namely X. This follows from the fact that if A e9l, then A cz X e9l.
Hence, if X cz A e 91, then A = X. This means that there is n o A e 9\
such that Α φ Xand X cz A, which proves that Xis a maximal element.
N o set A e 91 other than Ζ is a maximal element, since A cz X.
(II) There are no maximal elements in the ordered set (Jf, |) (cf. Example (II), Section 1), since n\2n and η φ In for every η eJf, so that
η < In, and In eJf
(III) There are no maximal elements in the ordered set (âl®, <)
(cf. example (III), Section 1). F o r let / be any function in 91*. N o w
m
set g(x) = f(x) + l for every xe 0t. The function g is in 3l and the
formulas / < g and f φ g hold. Hence f <g. Consequently, / is not
a maximal element.
IX. ORDERED SETS
126
(IV) Consider now one by one the examples of finite ordered sets
given in Example (V), Section 1. In the first example (Fig. 11), the ordered set ({a, b}9
) has exactly one maximal element, namely b. In the
second example (Fig. 12), the ordered set ({a9 b, c}9 <) has exactly two
maximal elements, a and c. In the third example (Fig. 13), the ordered
) has exactly one maximal element, b. In the fourth
set ({a9b9c}9
example (Fig. 14), the ordered set ({a, b, c9 d}9
) has exactly one maximal element c. In the fifth example (Fig. 15), the ordered set
({al9
ai,ci?>,aâ>, # 5 , ci6}9
) has exactly two maximal elements, a5 and a6.
(V) Let X be the set of all natural numbers greater than 1, and let
be a relation on X defined thus:
y if and only if y\x.
x
It can easily be verified that the relation
on X defined in this way
orders the set X. In the ordered set (X, < ) there are infinitely many
maximal elements. For let ρ eX be any prime number. If n eX and
/7 Φ /?, then ρ
η does not hold, as η is not a divisor of p. Hence ρ
does not precede any element in X and thus is a maximal element. It
has been shown that every prime number ρ is a maximal element in X,
and hence X has infinitely many maximal elements.
It follows from the examples given above that in an ordered set there
may be one or more maximal elements, and even infinitely many such
elements, but also that there are ordered sets such in which there are
no maximal elements.
) is called the greatest
An element x0 e X of an ordered set (X9
element if the following condition is satisfied:
(3)
χ < x0
for every χ
eX.
It will be proved that
2.1. In an ordered set (X9 < ) there is at most one greatest
The greatest element is maximal.
element.
Assume that the following conditions are satisfied for every x e X:
χ
x0
and
χ
JC 0% where x0 eX
and x$
eX.
It follows that x%
x0 (substituting
for χ in the first formula) and
r
** (substituting x0 for χ in the second formula). As : is an antix0
2. MAXIMAL AND MINIMAL ELEMENTS
127
symmetric relation, this yields x0 = xg. This proves the first part of
Theorem 2.1.
). Should, for
Assume now that x0 is the greatest element in (X,
some χ eX, the conditions x0 < x and x0 φ χ be satisfied, then, as
χ <; x 0 , we would have χ = x 0 , which contradicts x0 Φ χ. We have
thus shown that x0 is a maximal element, which proves the second
part of Theorem 2.1.
Examples
In the ordered set (i)'i, cz) (Example (I)) the set X is the greatest
element. In the ordered set (yT, |) (Example (II)) there are no maximal
elements and hence, by 2.1, there is no greatest element either. Likewise,
in the ordered set (β®, <;) (Example (III)) there is no greatest element
as there are no maximal elements. Consider now, one by one, the finite
ordered sets discussed in Example (IV). In the first case (Section 1, (V),
Fig. 11) b is the greatest element. In the second case (Section 1, (V),
Fig. 12) there is no greatest element. In the third case (Section 1, (V),
Fig. 13) b is the greatest element. In the fourth case (Section 1, (V),
Fig. 14) c is the greatest element. In the fifth case (Section 1, (V), Fig. 15)
there is no greatest element. In the ordered set (X, < ) (Example (V))
there is n o greatest element, for should there be a greatest element x0
in this set, then, for every x e X, the formula χ < x0 would hold, so
that x 0 would be a divisor of every number x e X. Among the natural
numbers greater than 1 there is no number x0 that has such a property.
An element x0 eX of an ordered set (X, ^ ) is called minimal if it is
not preceded by any element of this set, i.e., if there is no χ e X such
that χ -< x0. In symbols:
(4)
~V
(x<*o).
xeX
In accordance with the definition of the relation -< (see Section 1, formula (1)), condition (4) may be written as follows without resorting
to the symbol -<:
(5)
~\/
(x C x0 AX0 Φ Χ).
xeX
Examples
(VI) Let 9 v be any non-empty family of subsets of a set Χ Φ 0 , such
that 9 ? contains the empty set O. The family 91 is ordered by the in-
128
IX. ORDERED SETS
elusion relation c . In the ordered set (91, cz) there is exactly one minimal element, namely O. The condition A cz Ο and Α φ Ο is not satisfied
by any A e 91, which proves that Ο is a minimal element. At the same
time, for every A e9l and Α Φ Ο, Ο cz A holds; hence, Ο <A, which
shows that A is not a minimal element.
(VII) In the ordered set (Jf, |) (Example (II), Section 1) there is
exactly one minimal element, namely 1. This is so because there is no
natural number η φ 1 such that n\\, which shows that 1 is a minimal
element; at the same time, the condition \ \n is satisfied by every natural
number η φ 1, so that I <n, which proves that η is not a minimal
element.
(VIII) In the ordered set (J^ — {1}, |) there are infinitely many minimal elements. For let ρ eJf — {1} be any prime number. F o r every
element η eJf — {1}, if η φ ρ, then the formula n\p does not hold. Hence
it follows that ρ is a minimal element. We have demonstrated that every
prime number is a minimal element in the set under consideration, and
hence this set has infinitely many minimal elements.
(IX) In the ordered set
Ο (Example (III), Section 1) there are
a
n o minimal elements. Let / b e any function in 9t . Set g(x) = f(x) — l
st
for every x e l . The function g is in 31 and the formulas g < / and
g Φ f hold. Hence g <f, which proves that / is not a minimal element.
(X) Consider now, one by one, the ordered finite sets discussed in
Example (V), Section 1. In the first case (Fig. 11) the ordered set ({a, b],
) has exactly one minimal element, namely a. In the second case
(Fig. 12), the ordered set ({a,b,c},
<) also has exactly one minimal
element, namely b. In the third case (Fig. 13), the ordered set ({a, b, c),
-..') has exactly two minimal elements, namely a and c. In the fourth
case (Fig. 14), the ordered set ({a, b, c, d), :) has exactly one minimal
element, namely a. Finally, in the fifth case (Fig. 15), the ordered set
({a1} a2, a3, a4, a5, a6}, < ) has exactly two minimal elements, namely
a1 and a2.
These examples show that in an ordered set there may be one or
more minimal elements, and even infinitely many minimal elements, but
sometimes there may be n o minimal element at all.
An element x0 eXof an ordered set (X, < ) is called the least element
3. SUBSETS OF ORDERED SETS
129
if the following condition is satisfied:
(4)
x0 < x
for every χ
eX.
As in the case of Theorem 2.1 we may similarly show that
2.2. In an ordered set (X, <) there may be at most one least
The least element is minimal.
element.
The easy proof of this theorem is left to the reader as an exercise.
Examples
In the ordered set (91, cz) (Example (VI)) the empty set Ο is the
least element. In the ordered set (JV, |) (Example (II), Section 1) the
number 1 is the least element. In the ordered set («yT —{1}, |) (Example
(VIII)) there is n o least element as n o natural number η Φ 1 is a divisor
of every natural number other than 1. In the ordered set (at®, <)
(Example (IX)) there are no minimal elements, and hence, by 2.2, there
is n o least element. Consider also the finite ordered sets discussed in
Example (V), Section 1. In the first case (Fig. 11) α is the least element.
In the second case (Fig. 12) b is the least element. In the third case
(Fig. 13) there is no least element. In the fourth case (Fig. 14) a is the
least element. In the fifth case (Fig. 15) there is n o least element.
3 Subsets of ordered sets. The Kuratowski-Zorn lemma
Let (Χ, Ο be any ordered set, and let A be a subset of X. Let <^\A
be the binary relation on A defined t h u s : for every x,y e A,
(1)
χ
\A y if and only if χ <^y.
The relation ^\A is called a relation < restricted to A. It can easily be
proved that
3.1. If (Χ, Ο
an ordered set.
is an ordered set and if A cz X, then (A, ^\A)
is also
It follows from the definition of the relation < | Λ and from the
assumption that < is an ordering relation, that, for every x, ye A,
the following conditions are satisfied:
IX. ORDERED SETS
130
Thus the relation
((x
\Ay)A(y
\Az))=>
((x
\Ay)A(y
\Ax))
(x
\Az),
^ (x = y).
\A orders the set A.
Usually, since there is no risk of misunderstanding, {A,
\A) is
written briefly (A,
), it being understood that what is meant is the
relation
restricted to A.
Examples
(I) Consider the ordered set {Jf, |) (Example (II), Section 1). Let A
be the set of all even natural numbers. The relation of divisibility restricted to A is also an ordering relation. (A, |) is an ordered set.
(II) Let X = {al9 a2, a3, a 4 , a5, a6}; consider the ordered set (X,
)
described in Example (V), Section 1, Fig. 15. Let A = {ax, a2, a3, a4]
and let Β = {a4, a5, a6}.
The diagrams of the relations -\A
are shown in Figs. 16 and 17. By 3.1, {A, χ ) and (B,
and C^B
) are ordered
sets.
Fig. 16
Fig. 17
A subset A cz X of an ordered set (X, < ) is called a chain if for every
pair x, y of elements of A
(2)
χ C j
or
j <
χ.
Examples
(III) The set {2, 4 , 8, 16, . . . , 2", ...} is a chain in the ordered set
(Jf, I) (Example (II), Section 1), since for every pair of those natural
m
numbers which are in this set, i.e., numbers of the form 2" and 2
m
m
(n, m eJf), the condition 2"|2 or 2 | 2 " is satisfied. The subset {ax, a3,
tf4, a5} of the ordered set (X, < ) described in Example (V), Section 1,
3. SUBSETS OF ORDERED SETS
131
Fig. 15, is a chain, since for every pair at, a} of elements of that set
at. On the other hand,
(/ = 1, 3 , 4 , 5, j = 1, 3 , 4, 5) at . a} or a}
the subset {a1,a2,a3}
is not a chain, as neither at <^a2 nor a2
αλ
holds.
Let A c l b e a subset of an ordered set (X, ^ ). An element x0 Ε X
is called an upper bound of the set A if, for every χ Ε A, the condition
(3)
χ
x0
is satisfied. An element x0 Ε X is called a lower bound of the set A if,
for every χ Ε A, the condition
(4)
*o
x
is satisfied.
Examples
(IV) Consider the ordered set (Jf, |) (Example (II), Section 1). Let
A = {12, 18, 30}. The natural numbers 1, 2, 3, 6 are lower bounds of
A, since each of these numbers is a divisor of the numbers 12, 18, and
30. The numbers 180, 360, 540, . . . , « · 180, ... are upper bounds of A,
since each of those numbers which are elements of A is a divisor of
these numbers.
(V) Let 91 be the family of all subsets of any set Χ Φ Ο. Consider
the ordered set (91, cz) (see Example (I), Section 1). Let (At)teT
be any
indexed family of subsets of X, and hence of elements of 91. Now,
U At is an upper bound of this family of sets, since At cz [J At for
teT
teT
every t e Τ (see Theorem 2.1 in Chapter IV). Moreover, each set Α Ε 9\
such which contains that union is also an upper bound of the family
(A )teT
t
of sets. The intersection
P|
teT
A and every set A Ε 91 contained
t
of sets, since
in this intersection is a lower bound of the family (A )
t teT
Ο At cz At for every t Ε Τ (see Theorem 2.1 in Chapter IV).
teT
The theorem known as the Kuratowski-Zorn lemma is used in proofs
of many mathematical theorems. The formulation of that lemma requires reference to concepts introduced in this chapter. That theorem
was first proved by K. Kuratowski
and its importance in applicaJ
) K. Kuratowski, Fundamenta Mathematicae 3 (1922), p. 89.
IX. ORDERED SETS
132
1
tions was first demonstrated by M. Zorn ) . The proof of this theorem
requires reference to the axiom of choice or to theorems the proofs of
which refer in turn to the axiom of choice. We shall, for the time being,
confine ourselves to formulating the Kuratowski-Zorn lemma, a proof
of which will be given in Chapter XI.
3.2 (THE K U R A T O W S K I - Z O R N LEMMA). Let (X, <) be an ordered set.
If
for every chain A cz X, there is an upper bound in X, then X has a maximal
element. More precisely, for every x0 e X there is a maximal element χ
such that x0 - ". χ.
2
4. Note on lattices )
Let {Χ, Ο be an ordered set, and let A cz X be any subset of X.
Consider the set of all upper bounds of A (see Section 3). It is a subset
of X, and hence, by 3 . 1 , it is ordered by the relation < restricted to
that subset. If the set of all upper bounds of A has a least element, then
this element is called the least upper bound of A and denoted by sup A.
Likewise, if the set of all lower bounds of A has a greatest element,
then this element is called the greatest lower bound of A and denoted
by inf A.
Examples
(I) Consider the ordered set {JV , |) and let A = { 1 2 , 1 8 , 3 0 } (see
Example (IV), Section 3). The set of all upper bounds of A consists of
those natural numbers which are common multiples of the numbers
which are elements of A. They are numbers of the form Λ: · 180, k
= 1 , 2 , . . . The set of those numbers is ordered by the divisibility relation I restricted to that set. The number 180 is the least element of this
set since, for every k = 1, 2 ,
180 | Â>180. Hence, sup A = 1 8 0 . The
set of all lower bounds of A consists of all common divisors of those
') M. Zorn, Bulletin of the American Mathematical Society 41 (1936), pp.
667-670.
2
) Lattice theory is treated extensively by G. Birkhoff in Lattice theory, American Mathematical Society Colloquium Publications XXV, New York 1940, 2nd
ed. 1948.
5. QUASI-ORDERING RELATIONS
133
numbers which are elements of A, and hence this is the set {1, 2 , 3 , 6}.
This set is ordered by the divisibility relation | restricted to it. The
number 6 is the greatest element of that set, since each of the numbers
1, 2, 3, 6 is a divisor of 6. Hence, inf A = 6.
(II) Consider now Example (V), Section 3. It follows from Theorem
2.1 (1), (2), Chapter IV, that
sup(At)teT
Likewise, it follows from Theorem 2.1 (1), (3), Chapter IV, that
The case of those ordered sets in which each pair of elements has
a least upper bound and a greatest lower bound are of special importance. Those ordered sets for which this condition is satisfied are
called lattices. A m o n g the examples of finite ordered sets given in Section 1 (Example (V)) the ordered set ({a,b}, < ) , Fig. 11, is a lattice,
and so is the ordered set ({a, b, c, d}, < ) , Fig. 14. The ordered sets
(91, <=) (Example (I), Section 1), where 91 is a non-empty family of
subsets of a set Χ φ Ο and satisfies the condition
if A G 9i and Β e 91, then A uB e 91 and A nB e 91,
are important examples of lattices. In such a case, for each pair A, Β of
elements of 91 there is a least upper bound (the union A KJB) and a greatest lower bound (the intersection A nB).
There is a rich and rapidly developing mathematical theory concerned
with lattices; it finds numerous applications in the various branches of
mathematics.
5. Quasi-ordering relations
A relation ρ c Xx X is called quasi-ordering on X if it is reflexive and
transitive, i.e., if the following conditions are satisfied for any elements
x, y, ζ of X:
(r)
(t)
XQX,
(χ ρ yAy
ρ ζ)
=> χ ρ
z.
134
IX. ORDERED SETS
If a relation ρ c l x l i s
a quasi-ordering on X, then we also say
that ρ quasi-or ders X, and the ordered pair (Χ, ρ) is called a quasiordered set.
Examples
(I) Let 2£ be the set of all integers and let ρ be a relation on J f defined t h u s : for any m, η in J f ,
m ρηο
\J(mk
= ή).
It can easily be verified that the relation ρ thus defined quasi-orders the
set
2.
Λ
(II) Consider the set 0ί " of all sequences with terms that are real
numbers. Let ρ be a binary relation on this set defined t h u s :
( t f „ W e ( 6 , , W o \ J /\(m
< n=> an
It can easily be verified that ρ quasi-orders
bn).
-
Λ
01 \
+
(III) Let X be the set of all f u n c t i o n s / : 01+ -> 0t, where 0t is the
set of all non-negative real numbers, and 01 is the set of all real numbers.
A binary relation ρ on Ζ is defined t h u s :
fQ g ο
V
Λθ
<
x
=>Ax) < g(x)) ·
It can easily be verified that ρ quasi-orders X.
Every quasi-ordering relation determines an equivalence relation and
1
an ordering relation in a natural way. The following theorem h o l d s ) :
5 . 1 . Let (Χ, ρ) be a quasi-ordered set, and let Ä be a binary
on X defined thus :
(1)
x>xyo
relation
(χ ρ y Ay ρ x).
The relation « is an equivalence relation on X. Let, for every χ and y in X,
(2)
\\x\\^\\y\\oxQy-
The binary relation ^ defined on the set Xj κ, (the set of all equivalence
classes of the relation « on X) by formula ( 2 ) orders the set Xj » .
*) This theorem was proved by E. Schröder in 1890.
6. DIRECTED SETS
135
The reflexivity of « follows from Definition (1) and from the reflexivity of ρ. The symmetry of « follows directly from Definition (1). The
transitivity of » is a consequence of the transitivity of ρ. Hence, « is
an equivalence relation on X.
If χ ρ y and also # « * i and y « yx, then it follows from the transitivity of ρ that Χ ι ρ . Thus, Definition (2) of the relation < is correct,
since, for every a = \\x\\ eX/π
and for every b = \\y\\ e Χ / π , the fact
whether a < b holds does not depend on the choice of the representatives
x, y of the classes a and b. The reflexivity and transitivity of < follow
from t h e reflexivity and transitivity of ρ. If \\x\\ < \\y\\ and ||^|| <
then χ ρ y and y ρ χ, and hence, by (1), χ « y. This and Theorem 1.1,
Chapter VII, yield ||x|| = \\y\\. Thus, < is antisymmetric and hence
orders the set X/& .
The construction of the ordering relation as described in Theorem
5.1 finds numerous applications in mathematics.
J
6. Note on directed sets )
A directed set of indices is an ordered pair ( Γ , ρ), where Τ is any set
and ρ is a binary relation on Τ which is transitive and also satisfies the
following Moore-Smith condition :
(M)
f \ / \ \ /
xeT yeT
{XQZ^QZ).
zeT
The formula χργ is then read: y is a successor of x. Condition (M)
states that for any pair x, y of elements of Τ there is a common successor.
Example
Let 91 be the set of all finite sequences whose terms are in the set
{0, 1}. A relation ρ on 9^ is introduced t h u s : for any (αλ,
l
) The concept of a directed set was formulated by Ε. H. Moore in 1915. Directed
sets find applications in various branches of mathematics and are investigated in
the theory of Moore-Smith convergence. The concept of a limit of a sequence and
of a function cannot be formulated generally without making use of this concept.
IX. ORDERED SETS
136
(&,,..., o ) IN 9 Î ,
m
(β!, ..., a j ρ ( Ô ! , ..., ftm) ο ( m < w Λ Λ 0
.Λ
n=>ak
=
bk)).
It can easily be checked that ρ is a transitive relation on 91 and that it
satisfies condition (M). Hence (91, ρ) is a directed set of indices.
Now let Χ φ Ο be any space, and let (Τ, ρ) be any directed set of
indices. Then a function / : Τ -» X will be called a directed set of elements
of the space X. It is usual to denote / b y (xt)teT, where xt = f(t) for
every r e Γ .
Example
Let the interval { x e ^ 2 : 0 < x < l } b e taken as X and let (91, ρ) be
the directed set of indices defined in the previous example. We set now,
an) e9l,
for every a = (ax,
xa = *(*!,...,«„) = * i / 2 + ..·
+aJ2\
T h e function (xa)ae& is a directed set of elements of the interval { x e l :
0 < χ < 1}.
The concept of a directed set of elements (points) of a space is
a generalized concept of a sequence of elements (points) of a space. It
finds applications in abstract branches of mathematics, above all in
topology, where it proves necessary t o generalize the concept of convergence of a sequence of points of a space so as to cover the case of
directed sets of points.
Exercises
T= { 1 , 2 , 3 , 4 , 5 , 6 } , At = {zeW:
1. Given a family of sets (At)teT,
< /} for every re Γ, where # is the set of all complex numbers, and
2
0
kt = \
|-1
, 1
\z-kt\
for / = 1,
for f = 2, 3 , 6 , '
f o r / = 4,
f o r / = 5,
determine whether the ordered set ((At)teT,
greatest, least.
c
) has elements: maximal, minimal,
J 37
EXERCISES
2. Let SF be the set of all those functions which map the set {1, 2, 3} into the
5
set {0, 1}. A binary relation ρ is defined on 3F thus: for every / , g in J ",
= / ( 0 for every i e { 1 , 2 , 3}).
fQgo(f(i)g<J)
Prove that ρ orders the set 3*. Indicate the maximal and the minimal elements in
( ^ , <?).
3. Let X be the set of all sequences whose terms are real numbers, and let ρ be
a binary relation on X defined thus: for any sequences (an)„ejr and (bn)nejr9
(an)nejr
Q(bn)neJT
o \ J
/\(k
< η => an <
bn).
Determine whether the relation ρ orders the set X.
4. Let X = { 1 , 2 , 3 , 4 , 5 , 6 , 7 } . A binary relation ρ on X is defined by the following formula: for any elements x, y of X,
χρ
yo2\x—y.
Show that ρ orders X. Draw a diagram of that relation and indicate the maximal
and minimal elements.
5. Give an example of an ordered set which contains:
(a) A:+1 elements, of which k are maximal, and 1 is minimal;
(b) k elements all of which are both maximal and minimal.
6. Give an example of an infinite ordered set that has :
(a) k maximal elements and infinitely many minimal ones;
(b) one minimal element, and all the remaining maximal ones.
7. What is the number of relations such that each of them orders a set of η
elements?
8. In an ordered set (X, < ) a ternary relation ρ is defined thus:
ρ(α, χ, b)ο (a < χ < b\/b < χ < a).
Prove that
the condition o(a, x, b) implies ρ(6, x, a);
the conditions ρ (λ, X, b) and ρ (α, b, χ) imply χ = b;
the conditions ρ (α, χ, b) and ρ(α, y, χ) imply
Q(a,y9b);
the conditions ρ(α, χ, b), ρ(χ, b9 y) and χ φ b imply ρ(α, b, y);
ç(b,c,d).
the conditions ρ(α, b9c) and ρ(α,ο,α) imply
9. Define a binary relation on the set # of all complex numbers which is an ordering of
10. Prove that in every non-empty finite subset A of an ordered set (X, < ) there
is a minimal and a maximal element. If A is a chain, then it also has a greatest and
a least element.
CHAPTER Χ
LINEARLY O R D E R E D S E T S
1. Linear orderings
A binary relation < on ^ which orders X (see Chapter IX, Section 1)
1
is called a linear ordering ) if it satisfies the following connectivity
condition :
(c)
X'f-.y
or
y
χ
for every x, y e X.
Thus the linear orderings on a given set X are reflexive, transitive, antisymmetric, and connected in X.
If a relation
is a linear ordering on X, then it is said that ^ linearly
orders X, and the ordered pair (X, < ) is then called a linearly ordered
2
set ) or a chain.
If (X, : ) is a linearly ordered set, then, under the convention adopted
in Chapter IX, Section 1, p . 124 we say that χ precedes y if χ < y and
χ Φ y, i.e., if χ -< y, where -< is the relation on X defined in Chapter IX,
Section 1, formula (1). It can easily be seen that if such is the case, then
the following condition is satisfied for any elements x, y of X:
(1)
if χ φ y, then χ -< y or y -< χ;
this condition follows directly from (c) and from the definition of the
relation -< on X.
Examples
(I) Let 0to be any subset of the set 01 of all real numbers. The relation "not greater t h a n "
on 0tQ linearly orders that set. Likewise, the
*) Linear ordering relations are called ordering relations by some authors. Ordering relations, in the sense of the definition adopted in the present book, are called
by them partial ordering relations.
2
) This concept is due to G. Cantor (1895).
1. LINEAR ORDERINGS
139
relation "not less t h a n " > on 0tQ orders that set linearly. Thus, ( ^ 0 ,
and (MQ, » are examples of linearly ordered sets, i.e., chains.
ζ)
(II) Let # be the set of all complex numbers, and let < be a binary
relation on # denned t h u s : for any (xt, yx) and (x2, y2) in
(2)
((χι,
yù
(χ2
> y2))
o
(fa
<
χ2)
ν ((χι
=
χ2)
Λ
> > 2) ) ) .
By (2), the relation ^ holds between two complex numbers (*i,
and
(JC 2 , y2) if and only if the real part χγ of the first number is less than
the real part x2 of the other number, or if the real parts of both numbers
are equal and the imaginary part y1 of the first number is not greater
than the imaginary part y2 of the other number. It can easily be checked
that the relation < thus defined on # linearly orders that set. Thus,
is a chain, where # is the set of all complex numbers and ^ is
the relation on # defined by formula (2).
(III) Let A be the set of all terms of the sequence (an)nçjr,
where
an = 3" for every η eJf. Then (A, |), where | is the divisibility relation
on A (so that n\m, if η is a divisor of m for any natural numbers n, m
in A), is a chain.
(IV) Let Τ be the set of all positive real numbers; assume that, for
every teT, At = {z e # : |z| < t}, where # is the set of all complex
numbers. This defines an indexed family of subsets of
We already
know (see Example (I), Chapter IX, Section 1) that the family 91
of all sets At91 e T, is ordered by the inclusion relation. It can easily
be verified that the inclusion relation as holding in the family 91 under
consideration linearly orders 9?. Hence, (91, cz) is a chain. It is, however, to be emphasized that usually the inclusion relation merely orders
families of sets.
The following theorem is easy to prove:
1.1. If (X,
) is a linearly ordered set, and if A cz X, then (Α, χ M)
is also a linearly ordered set.
By Theorem 3.1, Chapter IX, (A, <\A) is an ordered set. As condition (c) is satisfied in X, it is a fortiori satisfied in A, which proves that
r
the relation c\A is connected and that (A, \Ä) is a linearly ordered
set.
Under the convention adopted in Chapter IX, Section 2, an element
140
X. LINEARLY ORDERED SETS
j c 0 e X of a linearly ordered set ( Z , < ) is called the greatest element if
(3)
x < *o
for every x e l .
Instead of "the greatest element" we shall also say "the last element".
1.2. For every element xQ of linearly ordered set (X, <) the following
conditions are equivalent:
(4)
x0 is the last
element,
(5)
x < xQ for every x e X-
(6)
x0 is a maximal element, that is, ~\J((x
{x0},
φ x0) Λ (X0 <
x)).
xeX
Conditions (4) and (5) are equivalent, for if x0 is the last element,
then, for every χ eX, χ < x0; hence, the condition χ -< x0 is satisfied
for every χ eX—{x0}
(i.e., for χ eX a n d χ φ x0). Conversely, if the
condition χ - < JC 0 is satisfied for every x e X— {x0}, so that χ φ x0 and
x < * o , then, since we also have x0 -< x0, we infer that (3) holds for
every x e X, so that x0 is the last element. In proving the equivalence of
(4) and (5) we did not refer to the fact that < is a connected relation.
Thus these conditions are also equivalent for ordered sets. It was proved
in Chapter IX, Theorem 2.1, that the greatest (last) element is a maximal
element. Assume now that x0 is a maximal element. It follows that there
is no χ eX such that χ φ x0 and x0 <x. Thus, for every χ eX and
χ Φ x0, x0 ^.x does not hold. Hence and in view of (c), χ ^x0
holds
for every χ eX such that χ Φ x0. Since, at the same time, x0 < X o ,
hence the condition χ < x0 is satisfied for every χ eX. Thus, x0 is the
last element.
Under the convention adopted in Chapter IX, Section 2, an element
x0 eX of 3, linearly ordered set (X9 < ) is called the least element if
(7)
x0 · χ
for every χ
eX.
Instead of "the least element" we shall also say "the first element".
1.3. For every element x0 of a linearly ordered set (X, ^ ) the following conditions are equivalent:
(8)
x0 is the first element,
2. ISOMORPHISM OF LINEARLY ORDERED SETS
(9)
(10)
Xo <x for every x e Xx0 is a minimal element, that is, ~ \J
V
141
{x0},
((χ Φ
((*
* o ) a ( x < x 0) ) .
Conditions (8) and (9) are equivalent, for if x0 is the first element,
then, for every χ eX, x0 < x; hence, the condition x0 -< χ is satisfied
for every χ eX-{x0}
(i.e., for xeX and χ Φ x0). Conversely, if the
condition x0 < X is satisfied for every x e X- {x0}, so that χ φ x0 and
Xo < x> then, since we also have x0 < * 0 , it follows that ;c 0 < χ holds
for every χ eX9 so that x0 is the first element. In proving the equivalence of (8) and (9) we did not refer to the fact that < is a connected
relation. Thus these conditions are also equivalent for ordered sets. It
was proved in Chapter IX, Theorem 2.2, that the least (first) element is
a minimal element. Assume now that x0 is a minimal element. It follows
that there is n o x e X such that χ φ x0 and χ < x0. Thus, for every
χ eX and χ φ x0, χ < χ0 does not hold. Hence, in view of (c), x0 < χ
holds for every x e X such that χ Φ x0. Since x0 < x0, the condition
x0 < χ is satisfied for every χ eX9 which proves that x0 is the first
element.
1.4. If (X, <) is a linearly ordered set and if X is a non-empty finite
set, then (X, <) has a first and a last element.
If X is a one-element set, then the theorem is self-evident. Assume
now that the theorem holds for linearly ordered sets of η elements.
Assume that X = X0 u{a}9 where α φΧ0, and that X0 has η elements.
By 1.1, (X0, ^\X0) is a linearly ordered set. Since X0 has η elements,
hence, by the inductive assumption, (X09 ^.\X0) has a first element a±
and a last element an. If a < ai9 then a is the first element in (X9 <),
and if a± < a, then a^ is the first element in (X9 <.). One of these formulas
holds by formula (c) in Section 1. Likewise, we infer that if an < a9 then
a is the last element in the chain (X9 < ) , and if a < an9 then an is the
last element of this chain.
2. Isomorphism of linearly ordered sets
Two linearly ordered sets (X9 < ) and (X*9 < * ) are called similar or
isomorphic, which is denoted by the formula (X, <) ~ (X*, < * ) , if there
X. LINEARLY ORDERED SETS
142
is a one-to-one function / : X
X* that maps X onto X* and also satisfies the following condition for any x,y
eX:
A-
(1)
y if and only if f{x)
*/()')•
Examples
(I) Let J f be the set of all integers; consider ( J f , ), where - ; is the
relation "not greater t h a n " on Jf. Also let Jf be the set of all natural
numbers, and let .* be a relation on Jf which is defined t h u s : for any
m, η eJf,
m r * η if and only if m is an even number and η is an odd number,
or η < m and m, η are both even, or m
η and m , « are both odd.
The relation ;C* linearly o r d e r s ^ , and the linearly ordered sets ( J f ,
and {Jf', < * ) are isomorphic. For l e t / : J f -+Jf be a function which is
defined thus :
2fc + l
for*
>0,ke&,
Rk) = {-Ik
fork
< 0,
ke&.
The function / maps J f onto Jf, is one-to-one, and also satisfies the
condition: for any integers m,n,
(m < » ) o (/(m)
*/(«))·
Verification is left to the reader.
where an
(II) Let A be the set of all terms of a sequence (an)nejr,
= 1 — l/n for every n e N. The linearly ordered sets (Jf', < ) and (A, <)
are isomorphic. The function which establishes the isomorphism is / :
Jf -> A, defined by the formulaf(ri) = 1 - 1 \n for every η eJf.
The definition of isomorphism of linearly ordered sets implies that
2.1. If linearly ordered sets (X,
the sets X and X* are equipotent.
) and (X*,
*) are isomorphic, then
We now show that condition (1) can be replaced by a weaker one,
namely
2.2. The linearly ordered sets (X,
) and (X*, -<*) are isomorphic if
and only if there is a one-to-one function f: X -* X* which maps X onto
X* and also satisfies the following condition :
(2)
if x
y, then f (χ)
*f(y),for
every χ, ye
Χ.
2. ISOMORPHISM OF LINEARLY ORDERED SETS
143
Assume that / : X -+ X* is a one-to-one function, maps X onto X*
and satisfies condition (2). Assume also that the formula χ < y does not
hold for some elements x, y of X. This and the reflexivity of the relation
< yield that χ Φ y. We also infer from conditions (c), Section 1, that
s i n ce
/ i s a one-to-one function
y
x. This and (2) yield fly) < * / ( * ) ·
and χ φ y, h e n c e / Ο ) ^ /(x). Consequently, the formula flx) %*fly)
does not hold, since this formula, together with f(y) - * flx), would
imply / ( * ) = fly), which contradicts with /Ο) Φ flx). It has thus been
proved that if x
y does not hold, then flx) * fly) does not hold
either. This and condition (2) imply tUat condition (1) is satisfied for
the function / , which completes the proof of Theorem 2.2.
In the case of finite sets Theorem 2.1 can be inverted. It will be
proved that
2.3. If (X,
) and (X*,
ï = ï*,then(X,
.) ~ (X*,
*) are linearly ordered finite sets, and if
*).
Assume that X and X* have η elements each. If η = 1, then ^ is a
relation on ^ e q u a l toXxX, and likewise
is a relation on X* equal to
Χ* χ X*. A function / which associates with the unique element of X the
unique element of X* is one-to-one, maps X onto X* and also satisfies
condition (1), Section 1, and hence establishes the isomorphism of the
chains in question. Assume that the theorem holds for sets of η elements
each, and that X and X* are (n + l)-element sets. By 1.4, (X, < ) and
(X*, < * ) each has a first element. Let a be the first element in (X, <)
and let b be the first element in (X*,
Set A = X- {a} and Β = X* — {b}. A and Β have η elements each. By 1.1, (A, %\A) and (B, <*|2?)
are linearly ordered sets and are isomorphic by the inductive assumption. Hence there is a function / : A -> Β which is one-to-one, maps A
onto B, and satisfies condition (1) for every pair x, y of elements of A.
Define
{
f(x)
for every χ e A,
b
for χ = a.
The function g defined as above maps X onto X*, is a one-to-one function, and satisfies condition (1) for every pair x,y of elements of A.
Thus, g establishes an isomorphism of the chains (X, <".) and (X*, <*).
X. LINEARLY ORDERED SETS
144
2.4. The following formulas hold for any chains (X, <), (X*9^*)
(X, Ο
(3)
^
(X,
and
O ,
(4)
( ( * , < ) - (**, <,*)) => ((x*, x*) -
(5)
(((x, -
<)),
) ^ (**, χ*)) Λ ((**, χ*) ~ (*', <')))
= > ( ( X , 0 ~
:.')).
Formula (3), which states that every chain is isomorphic with itself,
holds because the identity function Ix (see Chapter III, Section 2,
formula (11)) is one-to-one, maps X onto X, and satisfies condition (2).
T o prove condition (4), which states that if the chains (X, t ) and
(X*, ^ *) are isomorphic, then (X*, ^ * ) and (X, ^ ) are isomorphic,
assume that the chains (X,
) and (X*, ^ * ) are isomorphic. Hence there
is a one-to-one function / that maps X onto X* and satisfies condition
(1). It follows from Theorems 2.3 and 2.2, Chapter III, that the inverse
1
function f" : X* -> X is one-to-one and maps X* onto X. Suppose that
JC*, y * e X* and that x*
y*. Since / maps X onto X*, hence there
are in X elements x, y such that x* = f(x) and y* = / ( y ) . Hence, the
formulas x*
* ^ * and (1) yield that χ < y. But the preceding equations
_ 1
1
and Theorem 2.2 (5), Chapter III, yield that χ = / ( * * ) and y = / " ( y * ) .
x
Consequently, f~ (x*) . / ^ ( y * ) - Hence and from 2.2 we infer that the
chains (X*9
*) and (X < ) are isomorphic.
T o prove (5) assume that the chains (X, < ) and (X*9 < * ) are isomorphic, and that the chains (X*9 ^ * ) and (X'9
are isomorphic.
It follows from the assumptions made above that there are two oneto-one functions, / : X
X*9 mapping X onto X*9 and g: X* -*X'9
mapping X* onto X\ which satisfy the conditions:
(6)
(7)
(x < j ) ο
*k*f(y))
(x* < * j * ) ο (g(x*) ί ξ ' g(y*))
for every X J G I ,
for every x*, y* e X*.
We infer from Theorem 3.3, Chapter III, that the function g of
X -> X' is one-to-one and maps X onto X'. Further, it follows from
(6) and (7) that if x
y, then g o f(x) ^' g of (y) for any elements
x, y of X. By 2.2, the chains (X9 χ) and (X'9 < ' ) are isomorphic.
Theorem 2.4 makes it possible to classify linearly ordered sets in
terms of the concept of isomorphism in the same way as Theorem 1.1,
2. ISOMORPHISM OF LINEARLY ORDERED SETS
145
Chapter VIII, permits the classification of sets in terms of the concept of
equipotence. T o every set we assigned the cardinal number or the power
of the set; analogously, in the case of linearly ordered sets we assign to
each such set (X, < ) the order type of that set, one and the same order
type being assigned with two linearly ordered sets if and only if those
sets are isomorphic. Hence we say of isomorphic linearly ordered sets
that they are of the same order type*).
By Theorem 2.3, every two finite linearly ordered sets of η elements
each are isomorphic, and hence of the same order type. The type of
finite linearly ordered sets of η elements each is denoted by n. The
type of the empty set is denoted by 0.
The following order types of infinite linearly ordered sets play a very
important role: the type of the set of all natural numbers, linearly
ordered by the relation "not greater t h a n " (denoted by ω ) ; the type
of the set of all negative integers, linearly ordered by the relation "not
greater t h a n " (denoted by ω*); the type of the set of all rational numbers,
linearly ordered by the relation "not greater t h a n " (denoted by η);
the type of the set of all real numbers, linearly ordered by the relation "not greater t h a n " (denoted by λ). N o t e also that the linearly
ordered sets of type ω are enumerable, have a first element, and have
n o last element; the linearly ordered sets of type ω* are enumerable,
have a last element and have n o first element; the linearly ordered
sets of type η are enumerable and have neither a first nor a last element;
the linearly ordered sets of type λ are of the power of the continuum
and have neither a first nor a last element. Other properties of the
linearly ordered sets of types η and λ will be discussed in sections that
follow.
We could dispense with order types, since the theorems of set theory
can be formulated in such a way that n o reference is made to the concept of order types: one uses instead the notion of isomorphism of
linearly ordered sets. But the introduction of the concept of order
types is convenient for many reasons not the least of which is its usefulness in formulating many theorems more clearly.
*) Order types are introduced into set theory by means of the axiom on the
existence of order types. See K. Kuratowski and A. Mostowski, Set theory,
Amsterdam-Warszawa 1967.
X. LINEARLY ORDERED SETS
146
3. Dense linear ordering
A linearly ordered set (X, C) is said to have a dense linear ordering
if, for every pair of elements x,y eX, the following condition is satisfied:
(1)
if χ < y, then there i s a z e l such that χ -< ζ -< y,
where -< is the relation on X defined in Chapter IX, Section 1, formula
(1). Condition (1) is written in symbols thus:
(2)
(*<y)=>
\J
(x<z<y).
zeX
For instance, the set Q of all rational numbers, linearly ordered
by the relation "not greater than", has a dense ordering, and so has the
set 0t of all real numbers, ordered by the relation "not greater than".
On the contrary, the set Jf of all natural numbers, linearly ordered by
the relation "not greater than", is not densely ordered.
3.1. If (X,
) has a dense linear ordering, and if (X*,
is a linearly
ordered set isomorphic with (X, < ) , then (Z*, *) also has a dense
linear ordering.
Let / : X -> X* be a function that establishes the isomorphism of the
given linearly ordered sets. Suppose that x* and j>* are any elements
of X* such that x* <* y*, i.e., such that x* <* y* and χ* Φ y*. It
follows therefrom that there are in X elements χ and y such that x*
= / ( x ) , y* = f(y), and χ <y (see Section 2, formula (1)). Since the
linear ordering < of X is dense, there is a z e X such that χ •< ζ -< y.
This implies that f(x) ·.*/(*)
* f(y) and f(x) φ {{ζ)ΦΑγ),
so that
x* -< z* -<
which proves that (X*, ^ *) has a dense linear ordering.
It follows from Theorem 3.1 that it makes sense to talk about dense
order types.
1
The following theorem, given here without p r o o f ) , characterizes
order type η.
3.2. Every enumerable set which is linearly ordered into a dense order
l
) The proof may be found in: F. Hausdorff, Set theory, Chelsea, New York
1957.
4. CONTINUOUS LINEAR ORDERINGS
147
type and which has neither a first nor a last element is isomorphic to
the set of all rational numbers ordered by the relation "not greater than",
and thus is of type η.
4. Continuous linear orderings
Let (X, < ) be a linearly ordered set, and let Xx and X2 be two subsets of X that satisfy the following conditions:
(1)
X1 uX2
= X,
(2)
X,nX2
= O,
(3)
if x1 ΕXx and x 2 e l 2 , then x1 •< x 2 .
A pair (Χλ, X2) of sets that satisfies conditions (1), (2), (3) is called
a cut of the set X. If the sets X1 and X2 are non-empty, then the cut
(X1, X2) is called a proper cut. The set X1 is called the lower class of
the cut, and the set X2, the upper class of the cut.
Example
Consider the set Q of all rational numbers, linearly ordered by
the relation < "not greater than". Let x1 be any rational number.
Set Xx = {xe Q: χ < x j and X2 = Q-Xx.
The pair (Xl9 X2) is
a cut in the linearly ordered set (Q, <).
If in a given cut (Xx, X2) the lower class has a last element and the
upper class has a first element, then it is said that that cut yields a jump.
For instance, in the set J f of all integers, ordered linearly by the relation "not greater than", every proper cut yields a j u m p . It follows
from the definition that every proper cut (Xx, X2) of the set of integers
is in the form X1 = {x e J f : χ < m}, X2 = {x e J f : m < χ } , where m
is an integer; hence w — 1 is the last element in the lower class Xl9 and
m is the first element in the upper class X2.
4.1. A linearly ordered set (X, ^) is of a dense type if and only if
no cut yields a jump.
Suppose that the set (Χ, Ο is ordered linearly into a dense type.
Let (Xx, X2) be any cut that yields a j u m p . Let x1 be the last element
in the lower class X1 and let x 2 be the first element in the upper class
X. LINEARLY ORDERED SETS
148
X2. We accordingly have x1 < x2. Hence there is an element χ eX
such that x1 < χ < x2. The element Λ: is not in X1, nor is it in X2,
which, however, is not possible, as X1 uX2 = X. We have thus proved
that if a set is ordered linearly into a dense type, then none of its cuts
yields a j u m p . Conversely, if a set X, ordered linearly by the relation
< , is not of a dense type, then there are elements x1 and x2 of X such
that x1 < x2 and such that there is in the set X n o element ζ
which satisfies the condition xx -< ζ -< x2. By setting X1 = {x e X:
χ < x j and X2 = {x e X: x2 < x} we define a cut (Χί, Z 2 ) of X such
that x x is the last element in X1 and x2 is the first element in X2. This
cut yields a j u m p .
If, in a cut (ΑΊ,Α^), the lower class X1 has n o last element and
the upper class X2 has n o first element, then we say that the cut yields
Si gap.
Example
Let X be the set of all rational numbers other than zero, ordered
by the relation "not greater than". Let X1 be the set of all negative
rational numbers, and X2, the set of all positive rational numbers.
The cut (Xi, X2) yields a gap.
A linearly ordered set (X9 •<) is said t o have a continuous linear
order if it has a dense order type and none of its proper cuts yields
a gap.
4.1. If(X, Ο has a continuous linear order and if(X*, <*) is a linearly
ordered set which is isomorphic with (X, <), then (X*9 <*) also has
a continuous linear order.
It follows from 3.1 that (X*, < * ) has a dense linear order. Let / :
X -» X* be a function which establishes an isomorphism of the two
linearly ordered sets in question. Also let (Xf ,Χξ) be any proper cut
-1
of Jf*. Then (ΓΗ**),/ ^*)) is a proper cut of X: we have
(4)
r w ^ r w )
= r w u j r * )
=r (x*)
i
i
= r ( / w ) =
χ
in view of Theorem 5.10, formula (18), Chapter VI, the definition
of a cut, and Theorem 5.11, formula (26), Chapter VI. Hence condition (1) is satisfied. Condition (2) is also satisfied, since
(5)
η
1
{XV) η / " (Χ$) = Γ
1
(Xt nXf)
1
= Γ
(Ο) = Ο
4. CONTINUOUS LINEAR ORDERINGS
149
in view of Theorem 5.10, formula (20), Chapter VI, the definition
of a cut, and the definition of an inverse image (formula (16) in Chapter
and
VI, Section 5). Condition (3) is satisfied as well, for if xl ef'^X*)
_ 1
x2 e / ( * 2 * ) , t h e n / O O e X ? and f(x2) e X * 9 and hence /(x,)
<*f(x2)
by the definition of a cut. Hence xx < x2. Suppose that the cut (Xf, Χξ)
yields a gap. If so, then Xf has no last element and Χξ has n o first
l
_ 1
has n o last element and / ( ^ * ) has
element. It follows that f' {Xf)
1
n o first element. F o r should χ Α ef~ (Xf)
be the last element in that
l
set, we would have /(x^) eXf and χ < χγ for every χ ef~ (Xf).
Hence
/(*) < * / ( * i ) would hold for every element f(x) of Xf. Since every
l
element of Xf is of the form f(x), where xef~ (Xf),
/ ( x j would
be the last element in Xf, which contradicts the assumption that X\
χ
has n o last element. We show in a similar manner that ί~ (Χξ)
has
n o first element. Thus the cut (/^(Xf),
/^(X*))
would yield a gap,
but this contradicts the assumption that X has a continuous linear
ordering.
It follows from Theorem 4.1 that the continuity of a linear ordering
of a set is a property which is preserved under .isomorphism. Thus
we may speak of continuous order types.
The set of all real numbers, ordered linearly by the relation "not
greater than", has a continuous order type.
This statement is just another formulation of Dedekind's principle
of continuity *), which is one of the axioms of the theory of real numbers.
The linear ordering of the set of all rational numbers by the relation "not greater t h a n " is not continuous. F o r instance, let X1 be
the set of all rational numbers less than j / 2 , and let X2 be the set of
all rational numbers greater than j / 2 . The cut (Χλ, X2)
of the set of
all rational numbers yields a gap.
If a set X, ordered linearly by the relation < into a dense type without a first and without a last element, has proper cuts which yield gaps,
then these gaps can be filled if new elements are joined to X by the
following procedure: if a cut (X1,X2)
yields a gap, a new element χ
J
) See
p. 18.
K.
Kuratowski,
Introduction
to
Calculus,
Oxford-Warszawa
1961,
X. LINEARLY ORDERED SETS
150
is joined to X, and at the same time the relation < is extended by the
adoption of the convention that x x < χ for every xx e Xx and χ <: x2
for every x2 eX2. If χ is an element joined to X by a cut
(Xl9X2),
and if x* is an element joined to X by a cut (Xf, Χξ), then the convention is adopted that χ < χ* if Xt cz Xf, and χ* < χ if
cz Xx. At
least one of these conditions is always satisfied, since otherwise we
φ Ο and Xf-X,
φ Ο (see Chapter I, Theorem 4.1,
would have Xt-Xf
formula (6)), and hence there would be in X elements y and ζ such
that y e X1 and y φ Xf, and ζ e Xf and ζ £ X1. That would yield :
y eXl and ζ e X—X1 = X2, so that y> •< z; at the same time this would
yield: zeXf
and yeX—Xf
=X2,
so that z<y.
These two conditions cannot both be satisfied, which proves that either X1 cz Xf or
Xf cz X1. It can be proved that by adding to X new elements determined
by all those proper cuts which yield gaps, we obtain a set, linearly ordered into a dense type without a first and without a last element,
such that none of its proper cuts gives a gap, and hence a set which
is linearly ordered into a continuous type.
Application of the above method of filling gaps to the set Q of
all rational numbers, linearly ordered by the relation "not greater
than", results in Dedekind's theory of irrational numbers
Exercises
1. Determine whether the relation o, defined in the set Λ"οϊ all natural numbers
thus:
χ ρ y o (x\y Vx < y),
linearly orders that set.
2. Determine whether the set ^Vof all natural numbers and the set of all numbers
2
in the form 2—l/// , ne-yV, are linearly ordered by the relation "not greater than"
in a similar way, i.e.. into the same order type.
3. Let (X, < ) be a linearly ordered set. If xe X, and if the set {y G X:
x<yA
Αχ φ y} has a first element, then that element is called the successor of x. If xeX
and if the set {y eX: y < XAX φ y} has a last element, then that element is called
the predecessor of x. Specify a relation which orders the set ^ o f all natural numbers
linearly so that each element has a predecessor and a successor.
4. Prove that a linearly ordered set (X, < ) is of type ω if and only if the following conditions are satisfied: 1° X has a first element, 2° every xeX has a successor
1
) See K. Kuratowski, op. cit., p. 23.
151
EXERCISES
in X9 3° if the first element of X is in A <= X and if A contains the successor of each
of its elements, then A — X.
5. We say that a family of sets is monotonie if it is linearly ordered by the inclusion relation. Prove that every linearly ordered set is isomorphic to a monotonie
family of sets.
Hint. For every element χ of a linearly ordered set (X, < ) under consideration, let P(x) = {yeX: y < χ Αχ Φ y}. Take (P(x))xex to be the corresponding
monotonie family of sets.
6. Let (X9 < ) and ( X * 9 < * ) be sets linearly ordered into type η, such that XnX*
= O. Set Y = XvX* and define in Y a binary relation ρ thus: for any x, y in Y9
x ρ y o((x
e XAy e X*)v (χ e XAy € XAX < y)v (x e X * Ay e X * AX
<*y)).
Prove that (Y9 ρ) is a set linearly ordered into type η.
7. Let (X, < ) and (X*9<*)
be sets linearly ordered into type ω and such
that XnX* = O. Set Y = XvX* and let ρ be a binary relation in Y9 defined as in
Exercise 6. Prove that (Κ, ρ) is a linearly ordered set whose order type is other than ω.
8. Prove that the set of all numbers in the form 1 ///, where n e 2— {0} (i2f being
the set of all integers) does not contain any infinite dense subset.
9. Let (X9 < ) be a linearly ordered set. A subset A <=• X is called dense in X if,
for every pair x9y οϊ elements of X9 such that χ < y and χ φ y, there is an a e A
such that χ < a < y and χ φ α φ y. Prove that every linearly ordered set (X9 <)
of type λ contains an enumerable subset which is dense in X.
10. Let # be the set of all complex numbers, and let < be a binary relation in
^ , defined thus: for any complex numbers z+iy and u+iv,
χΛ-iy < u-\-ivo(y
< w(y
= ν AX < u)).
Prove that
< ) is a linearly ordered set such that ^ does not contain any enumerable subset dense in ^ .
CHAPTER Xï
WELL-ORDERED SETS
1. Well-ordering relations. Ordinal numbers
A binary relation < o n I which establishes its linear ordering (see
Chapter X, Section 1) is called a well-ordering if it satisfies the following
condition :
(w)
for each non-empty subset A of the set X the linearly ordered
set (A, ^\A) has a first element.
If < is a well-ordering relation on X, then we also say that < wellorders X, and the ordered pair (X, < ) is called a well-ordered
set*).
2
The order types of well-ordered sets are called ordinal
numbers ).
The ordinal numbers include the order type of the empty set, which, under the convention adopted in Chapter X, Section 2, is denoted by 0.
Examples
(I) The set Jf of all natural numbers is well-ordered by the relation "not greater than". The type of the set of all natural numbers,
well-ordered by the relation "not greater than", which is denoted by
o) (see Chapter X, Section 2), is thus an ordinal number.
(II) It follows from Theorem 1.4, Chapter X, that if (Χ, Ο is a linearly ordered set, and if X is a finite non-empty set, then (X9 < ) is
well-ordered. Thus every finite set which is linearly ordered is wellordered. It is also known (see Chapter X, Theorem 2.3) that any two
«-élément finite sets which are linearly ordered are isomorphic.
It follows therefrom that the order types of finite linearly ordered
sets are ordinal numbers. Under the convention adopted in Chap*) The concept of well-ordered set was formulated by G. Cantor in 1883.
) The concept of ordinal number is due to G. Cantor in 1883.
2
1. WELL-ORDERING RELATIONS
153
ter X, Section 2, the ordinal number (order type) of a well-ordered
set of η elements is denoted by n. N o n e of the order types: ω*, η, λ is
an ordinal number.
The following theorem results immediately from Theorem
Chapter X, and from the definition of a well-ordered set:
1.1. If (Χ, Ο is a well-ordered
is also a well-ordered set.
1.1,
set and if A cz X, then (Α, < | Λ )
Let (Χ, Ο be a well-ordered set, and let x e X. If χ is not the last
element in X, then the set {y eX: (x <y) Α (Χ Φ y)} = {y eX: x <y)
is not-empty. Since this set is a subset of X, we infer by Theorem
1.1 that it is well-ordered and hence has a first element. It is called
the successor of x9 since it follows χ immediately. The definition of a
successor implies that
1.2. Every element χ of a well-ordered set, with the exception of the
last element, if it exists, has a successor.
N o t e that if x0 is the first element of a well-ordered set, then this
set has n o element χ such that χ -< x0, and hence x0 is not a successor
of any element of this set. A well-ordered set may have elements different from the first element which are not successors of any element.
Example
Let X be a set which consists of numbers in the form l — l/n (n
= 1 , 2 , . . . ) and the number 1. The set X is well-ordered by the relation "not greater than". The number 1 is not the first element of X,
nor is it a successor of any element of X.
Let (X, < ) be a well-ordered set. A subset Y cz X is called an initial
interval of X if the following condition is satisfied for every x,y
eX:
((yeY)A(x^y))^(xeY).
(1)
This condition states that whenever Y contains an element y it also
contains every element χ eX such that χ < y.
The following theorem will now be proved:
1.3. For every initial interval Y of a well-ordered
Υ φ X, there is an element y of X such that
set (X, <),
(2)
χ
Y = {xeX:
(x ^y)A(x
Φ y)}
= {x eX:
<y}.
where
XI. WELL-ORDERED SETS
154
The set X- 7 is a subset of X. Since, by assumption, Υ Φ X and
Y α X, hence Χ— Υ φ Ο (see Chapter I, Theorem 4.1, formula (6)).
Let y be the first element in X— Y. We show that y satisfies condition
(2). Assume that x e Y. Should y < χ, then, in view of the fact that Y
is an initial interval of X, y would be in Y, which is impossible as
y e X— Y. Hence y < χ does not hold. From this fact, and from the
connectivity of the relation < , we infer that χ <C y. Since χ Φ y (x e y,
and y φ 7 ) , we have shown that if x e Y, then χ < y and χ φ y. Conversely, assume that x ^y and χ φ y. It follows from this and the
assumption that y is the first element in X— Y, that χ φ X— Y. Hence
we infer that xeY (see Chapter I, Section 4, formula (2)). We have
thus proved that y satisfies condition (2).
Assume that, for every element y of a well-ordered set (X, <),
(3)
P(y)
=
{xeX:
(x
^y)A(x*y)}
=
{xeX:
x<y}.
It will be proved that the following formulas hold:
(4)
if y i
v , , then P(yt)
(5)
if j , Φ y2,
then P(yi)
<=/»(y 2 ),
Φ
P(j2)
for any elements y^, y2 of X.
Assume that yl < y2. It follows from the transitivity of the relation < that, for every x e Χ, χ < yt implies χ < y2. If χ e P(yi), then
x < j > i and χ Φ y γ. This fact and the previous conclusion imply
χ < y2. But χ = y2 cannot hold, for if yt = y2, then χ Φ y2 follows
from the condition χ Φ y±. If, on the other hand, yt Φ y2, then the
assumption that χ = y2 implies that y2 ^yi9
since χ < y{. Simultaneously we have yt < j 2 . This and the antisymmetry of the relation
< imply that y± = y2, which contradicts the assumption. We have
thus proved that if x G P(yi)9 then χ < y2 and χ φ y2, so that χ e P(y2).
The proof of formula (4) is thus complete.
Now assume that yr Φ y2. It follows from the connectivity of the
e n
relation ·. that either yl < y2 or y2 <
. If ^ < j>2» t h it can easily
W e
be seen that Ρ Ο ί ) # P(y2), as ^ e P ( y 2 ) and Ji ί ^ Ο ί ) ·
reason
similarly in the case y2 < yv. Formula (5) is thus proved.
The following theorem will now be proved:
1. WELL-ORDERING RELATIONS
155
1.4. Let (Χ, Ο be a well-ordered set, and let 91 be the family
all initial intervals of X other than X. Then
(i)
(ii)
the inclusion relation
cz linearly orders the family
of
91,
(X, <:) is isomorphic to (91, cz).
The inclusion relation orders every family of sets (see Chapter IX,
Section 1, Example (I)). T o prove (i) it suffices to demonstrate that the
relation cz is connected in the family 91. Let 7 X and Y2 be any initial
intervals of X other than X. It follows from 1.3 and (3) that there are
elements yt and y2 of X such that Yx = P(yJ and Y2 = P(y2). Since
< is a connected relation in X, either y1 < y2 or y2 < yx. The first
case implies P(y1) cz P(y2),
while the second implies P(y2) cz P(y1)
(by (4)). Hence it follows that either Yt cz Y2 or Y2 a Ylt Formula (i)
is thus proved.
Now set
f(y) = P(y)
for every y e X.
We have thus defined a mapping / : X -+91. It follows from 1.3 and
from (3) that every initial interval in 91 is in the form f(y) for the corresponding y eX. Thus / maps X onto 91. It follows from (5) that /
is a one-to-one function, and from (4), that if y± <j> 2 > then P(yi)
cz P(y2).
By referring to Theorem 2.2, Chapter X, we conclude that
(ii) holds.
It is easy to prove that
1.5. If(X, <) is a well-ordered set, and if(X*, C*) is a linearly ordered
set isomorphic to (X, <), then (X*, <-.*) also is a well-ordered set.
Let / : X-> X* be a function that establishes the isomorphism.
Let 7 * be any non-empty subset of X*. The inverse image of 7 * under
1
Thus Y cz X and
fis then a non-empty subset of X. Set Y =f~ (Y*)%
Υ Φ Ο. Y has a first element y; but y ef-^Y*)
if and only iff(y) e 7 * .
Hence f(y) is an element of Y*. It will be shown that f(y) is the first
element in 7*. Should this not be the case, there would be in 7 * an
element x* such that x* ^*f(y)
and χ* φ f(y). The element x* is in
the form f(x), where x e Y. Hence we would have, for an element
x e Y, f(x) ?Cf(y) and f(x) Φ f(y). As / establishes an isomorphism,
that would imply that χ <y and χ φ y. Hence y would not be the
156
XI. WELL-ORDERED SETS
first element in Y, which would contradict the assumption. The proof
is thus complete.
It follows immediately from Theorems 1.4 and 1.5 that
1.6. If (Χ, Ο is a well-ordered set, and if
is the family of all initial
intervals of X other than X, then the inclusion relation c
well-orders
the family 81, and (X, <) is isomorphic to (9ί, c z ) .
We also prove the following theorem:
1.7. Every function f: X-+X*
which establishes an isomorphism
between two well-ordered sets (X, <) and (X*, <*) maps any initial
interval of the former set onto an initial interval of the latter set.
Obviously / maps X onto X*. Thus we may confine ourselves t o
considering those initial intervals of Ζ which are other than X. It follows
from Theorem 1.3 that every such interval is in the form P(y) (see
(3)), where y is a corresponding element of X. We shall show that
f(P(y))
is an initial interval of X*. Set Ζ = {z e X*: ( Z < * / O 0 ) A
Α(ΖΦ f(y))}. Ζ is an initial interval of X*. We show that/(P(>>)) = Z .
If an element ζ of X* is in f(P(y)),
then there is an χ e P(y) such that
ζ — f(x). The element χ satisfies the conditions χ ^y
and χ φ y.
Hence we infer that f(x) </0>) and f(x) Φ f(y). Consequently, f(x)
e Z , i.e., ζ eZ. We have thus shown that f(P(y))
<= Z . If an element ζ
of X* is in Z , then ζ <*/0>) and ζ Φ f(y). The element ζ is of the form
f{x), where xeX.
This and the previous conditions imply that f(x)
<*/(>>) and f(x) Φ f(y). As / establishes an isomorphism, we may
infer that χ < y and χ Φ y. Thus x e P(y). Hence f(x) ef(P(y))9
that
is, zef(P(y)).
We have thus shown that Ζ cz f(P(y)),
which, taken
together with the inclusion already proved, shows that the sets in question are equal and concludes the proof of Theorem 1.7.
2. Comparison of ordinal numbers
Let (Χ, Ο be a well-ordered set of type a, and let (X*, < * ) be
a well-ordered set of type β. We say that the ordinal number α is less
than an ordinal number ß, in symbols
0)
«<β,
2. COMPARISON OF ORDINAL NUMBERS
157
if the set (X, < ) is isomorphic to an initial interval of the set (X*, <*)
distinct from (X*, < * ) . If this is the case then, obviously, every wellordered set of type α is isomorphic to an initial interval of any wellordered set od type β.
We shall write
(2)
instead of α < β or α = β.
The ordinal number 0 is less than any other ordinal number.
Example
Let α be the order type of a well-ordered set (Χ, Ο and let X*
= I u { û } , where α φΧ. Let < * be a binary relation on X * defined
t h u s : for any elements x, y of X*,
(3)
χ
y if and only if either y = a or the following conditions
are both satisfied: x,y eX and χ <>\
It follows from this definition that the relation < * holds between
elements of X if and only if the relation < holds. Moreover, for every
xeX*,
x^*a.
It can easily be proved that the relation < * wellorders X*. The order type (ordinal number) of the well-ordered set
(X*, < * ) will be denoted by α + 1 . We now show that
(4)
α < α + 1.
Χ = {x e Χ* : (χ
α) Λ (Χ Φ a)} under the definition of < * . Obviously,
Χ Φ X*. The identity function in X establishes an isomorphism of
the sets (X, < ) and (X, < * | X ) (i.e., the set X treated as an initial interval
of the set (X*, < * ) ) . Hence formula (4) holds.
We now show that
2.1. For every ordinal number a,
(5)
(α < α).
T o prove this theorem it suffices to show that n o well-ordered set is
isomorphic to any of its initial intervals other than itself.
Assume that / : X
P(y) maps a fixed well-ordered set (X, ^ ) onto
an initial interval P(y), where y eX, and that / establishes an isomorphism between these two sets. Notice that P(y) = {x eX: (x ^y)A
158
XI. WELL-ORDERED SETS
(x φ y)} (see Section 1, formula (3)) and that Theorem 1.3 states
that every initial interval other than X is of this form. Obviously, f(y)
G P(y). This and the definition of P(y) imply f(y) < y and alsof(y) Φ y.
Hence the set A = {x eX: (f(x) < Χ)Α(/(Χ)
Φ χ)} is non-empty. Consequently, this set has a first element. Let a be the first element of this
set. Then we have
A
(6)
(Λα)<α)Λ(/(α)Φ
a).
Since / establishes an isomorphism between X and P(y), it follows from
(6) that
(7)
( / ( / ( * ) ) <f(ßj)
A (f(f(a))
Φ
f(a)).
We infer from (7) that f(d) e A. Since condition (6) is also satisfied by
f(a),
the element a is n o t the first element of A, which contradicts
the assumption. The proof of Theorem 2.1 is thus complete.
2.2. For any ordinal numbers α, β , y ,
(8)
if (χ < β and β < γ, then a < γ.
Let oc, β , γ be, respectively, the order types of the following wellordered sets: (X, < ) , ( 7 ,
( Z , < 2 ) . By assumption there is a function / : X
Y which establishes an isomorphism between X and an
initial interval Y1 of Y different from Y and there is a function g:
Y -+ Ζ which establishes an isomorphism between Y and an initial
interval Zx of Z, different from Z. T h e composition of these two functions, g ο / , is a one-to-one function because / and g are both one-toone functions. Moreover, for any elements xt, x2 of X if xx < x 2 ? then
< i / ( * 2 ) and hence
< 2 g ( / f e ) ) , i.e., g o / f o ) < 2 g
of(x2).
The function / maps X onto an initial interval Υγ of Γ different from Y.
Thus / P O = Y1 φ Y. As the function g establishes an isomorphism
between Y and Zx, by Theorem 1.7 it maps the initial interval Yx of 7
onto an initial interval Z 2 of Z x . Hence it follows that g ο f(X) = g(J(X))
= g(Yx) = Z 2 . Obviously, Z 2 c Z , c Z , and Ζχ Φ Ζ , so that Z 2 φ Ζ .
Hence we infer that gof maps ^ o n t o an initial interval of Z, different
from Z. Consequently, g o / e s t a b l i s h e s an isomorphism between X and
an initial interval of Z, different from Z, which proves that oc < γ.
2. COMPARISON OF ORDINAL NUMBERS
159
It follows from 1.1 and 1.2 that
2.3. For any ordinal numbers α, β,
if α < β, then ~ (β < α).
(9)
Indeed, should α < β and β < α both hold, then, by Theorem 2.2,
we would have α < α, which is impossible in view of Theorem 2.1.
Consider now the proof of the following trichotomy theorem for
ordinal numbers:
2.4. For any ordinal numbers en, β, one of the following
holds :
a < β,
(10)
oc = β,
conditions
β < α.
T o prove this theorem we show that if (X, < ) and (Y, < J are any
well-ordered sets, then one of them is isomorphic to an initial interval
of the other. The following notation will be used: for every χ eX and
yeY,
let Px(x) and PY(y) be, respectively, the sets defined t h u s :
OD
Ρχ(χ)
= {ζ eX: (ζ <
(12)
P,{y)
=
{zeY:
*) Λ
(ζ Φ χ)},
(ζ < > > ) Λ (Ζ Φ
y)}.
By definition, the sets Px(x), where χ eX, and PY(y), where yeY,
are,
respectively, initial intervals of X and Y and are, in each case, different
from X and Y (see 1.3). Let Ζ be the set of those elements Λ: of I for
which there is an y e Y such that the interval Px(x) is isomorphic to
the interval PY(y). The definition of Ζ is written in symbols as follows:
(13)
Ζ = {xeX:
\J Px(x)
is isomorphic to
PY(y)\.
We may confine ourselves to the case where both sets are non-empty,
since if one of the ordinal numbers (order types) of these sets equals 0,
then the theorem holds. The set Ζ is non-empty, because it contains
the first element of X. N o t e that, in view of the fact that different initial intervals of a well-ordered set cannot be isomorphic (see the proof
of Theorem 2.1), for every xeX
there is at most one yeY
such
that Px(x) is isomorphic to PY(y). Hence there is a function f: Ζ -+ Y,
which may be defined by setting for every x e Z,
(14)
f(x) = ν if and only if Px(x)
is isomorphic to
PY(y).
160
XI. WELL-ORDERED SETS
We now show that Ζ is an initial interval of X. Assume that χ eZ
and that xt < x and x x φ χ. We show that x± eZ.
Since
xeZ,
the interval Px(x) is isomorphic to the interval PY(f(x)).
Hence
onto PY(f(x))
which establishes an
there is a mapping of Px(x)
isomorphism of these intervals. U n d e r this mapping, Px(Xi),
being an
initial interval of Px(x), is mapped onto an initial interval of
PY(f(x))
(see 1.7), and hence onto an initial interval of Y. It follows that
Px(xi)
is isomorphic to an initial interval of Z . This proves that xx e Z .
Likewise, the s e t / ( Z ) is an initial interval of Y. Assume that y e f(Z),
yx < y and yx Φ y. We show that y1 ef(Z). As y ef(Z), there is an
x G Ζ such that y = f(x). This and formula (14) imply that Px(x) is
isomorphic to PY{y). Hence there is a mapping of PY(y) onto Px(x)
which establishes an isomorphism between these two intervals. Under
this mapping, PY(yi), being an initial interval of PY(y), is mapped onto
an initial interval of Px(x), i.e., onto an initial interval of X. Hence,
Py(yù is isomorphic to an initial interval Px{Xi)
of X. Thus, for y^
there is an ^ e Z such that yx = /(x^. This proves that yt ef(Z). The
function / obviously maps Ζ onto f(Z). It is a one-to-one function,
which follows from the fact that no two distinct initial intervals of
a well-ordered set are isomorphic, and from the definition of / . It has
also been proved that if x± < x, then P y ( / ( * i ) ) is an initial interval of
Ργ(/(χ))' This proves that f(xx) < i / ( x ) (see 1.6). T h u s , / e s t a b l i s h e s an
isomorphism between Ζ and / ( Z ) .
It will be proved that Ζ = X or f(Z) = Y. Suppose that Ζ Φ X and
f(Z) Φ Y. Since Ζ is an initial interval of ^different from Χ, Ζ = Px(x)
for some χ eX (see 1.3). Likewise, since f(Z) is an initial interval of Y
different from Y,f(Z) = PY(y) for some y e Y. It follows from this fact
and from the isomorphism of Ζ and / ( Z ) that Px(x) is isomorphic to
PY(y). Thus it follows from the definition of Ζ that x e Z . Accordingly,
χ e Px(x),
which is impossible, since this would mean that χ < χ and
χ Φ χ. Hence Ζ = X or / ( Ζ ) = 7 . If Ζ = X , then Ζ is isomorphic to
an initial interval of Y, and if f(Z) = Y, then 7 is isomorphic to Z,
that is, to an initial interval of X. The proof of the theorem is complete.
The trichotomy theorem for ordinal numbers, Theorem 2.1, Chapter
X, and formula (2), Section 4, Chapter VIII, immediately yield the following theorem:
3. SETS OF ORDINAL NUMBERS
161
2.5. For any well-ordered sets (X, < ) and (Y, < ! ) , one of the following conditions holds :
(15)
X < Y,
X = Y,
Y < X.
It follows from Theorems 2.2, 2.3, 2.4, and formula (2) that
2.6. For any ordinal numbers α, /?, γ,
(16)
a<a,
(17)
if a < β and β < y , then a <
(18)
if a <
(19)
β and β <
a, ÎAÎW <x =
γ,
β,
a < j8 or β < a.
3. Sets of ordinal numbers
F o r every ordinal number a, let Z ( a ) denote the set of all ordinal
numbers less than a. It will be proved that
3.1. The set Z ( a ) is well-ordered into type a by the relation
<.
The set Z ( a ) is linearly ordered by the relation < by Theorem 2.6.
Let (X, •<) be a well-ordered set whose order type is the*ordinal number
a. Let / b e a function that with every element χ eX associates the order
type of the initial interval P(x) (see Section 1, formula (3)). Now, / maps
X onto Z ( a ) , for if β e Z(oc), then β < a. Hence it follows that every
well-ordered set whose order type is β is isomorphic to an initial interval P(x) of X (see 1.3). Hence β is the order type of P(x)9 which means
that f(x) = β, and this in turn shows that / m a p s X onto Z ( a ) . Moreover, it has been proved in Section 1, formulas (4) and (5), that if
Xi < * 2 > then Ρ(χχ) cz P(x2), and that the condition xx Φ x2 implies
P(xi) φ P(x2)> This leads to the following two conclusions. First, if
^ ! < ^ 2 ) then the order type of P(*i) is less than, or equal to, the order
< X 2 > then f(xi)
type of P(x2). Hence / satisfies the condition: if
^ / ( * 2 ) · Secondly, if xt Φ x2, then, since Pix^) φ P(x2)> and no two
different initial intervals of one and the same well-ordered set are isomorphic to one another, we may infer that the order type of P(*i)
differs from the order type of P(x2), so that f(xt) φ f(x2).
Thus, / is
a one-to-one function. By Theorem 2.2, Chapter X, the function /
establishes an isomorphism between the sets X and Z ( a ) . If follows
162
XT. WELL-ORDERED SETS
from this fact and from Theorem 1.5 that Z ( a ) is well-ordered by the
relation ^ and that its order type is a.
3.2. Every set of ordinal numbers is well-ordered by the relation
Let Ζ be any non-empty set of ordinal numbers. By Theorem 2.6,
the relation : orders Ζ linearly. Hence it suffices to show that every
non-empty subset Z x of Ζ contains a least number. Let α e Ζγ. If α is
not the least ordinal number in Zx, then the set Ζγ η Ζ ( α ) is non-empty.
As it is a subset of the well-ordered set Z ( a ) , it contains a least ordinal
number β. That number is also the least ordinal number in Z A , for if
γ e Zj - Ζ ( α ) , then α < γ and hence β < γ. Thus the ordinal number
and less
β is less than any ordinal number γ which is in Zlr\Z(ai),
than any ordinal number γ which is in Zx— Z ( a ) . Since Z x = ( Z x —
— Z ( a ) ) u ( Z 1 n Z ( a ) ) , β is less than any ordinal number of Z x .
Finally, it will be proved that
3.3. For every set Ζ of ordinal numbers, there is an ordinal number
which is greater than any ordinal number in Z.
Let Z * be the union of all sets Ζ(β), where β e Z . We may infer
from the definition of Z * that an ordinal number γ is in Z * if and only
if there is a β e Ζ such that γ < β (see Chapter IV, Section 1, formula
(1)). Hence it follows that if β e Z , then every ordinal number γ which
is less than β is in Z*. Consequently, for every β e Ζ, Ζ(β) is an initial
interval of Z * . If Ζ * = Ζ(β), then by Theorem 3.1
is the order type
of Z * . If, on the contrary, Ζ * Φ Ζ(β), then β is an ordinal number
which is less than the order type of Z*. Let the order type of Z * be
denoted by a. We have shown that β < α for every ordinal number β
which is in Z. Since α < α + l (see Section 2, formula (4), p . 157), this
fact and Theorem 2.2 imply that β < α + l for every ordinal number
ßeZ.
Thus α + l is an ordinal number which is greater than any
ordinal number that is in Z .
The following conclusion results from Theorem 3.3:
x
3.4. There is no set of all ordinal numbers ).
Thus it is not possible to form a set of all ordinal numbers, just as
it is not possible to form a set of all sets.
*) Before the formulation of axiomatic set theory this theorem was regarded
as a paradox. It was discovered by C. Burali-Forti in 1897.
4. POWERS OF ORDINAL NUMBERS
163
4. Powers of ordinal numbers. The cardinal number K(m)
The power of any well-ordered set whose order type is α is called the
power of the ordinal number a. Obviously, the power of a does not
depend on the choice of the well-ordered set, since any two isomorphic
well-ordered sets are of the same power (see Chapter X, Theorem 2.1).
For instance, the power of the ordinal number ω equals K 0 .
Since the set Z ( a ) of all ordinal numbers which are less than α is
well-ordered by the relation "not greater t h a n " into type α (see Theorem
3.1), the power of the ordinal number α equals Z ( a ) . If we denote the
power of the ordinal number α by α we have
(Ο
α = zfâ.
For every cardinal number m, let Z ( m ) denote the set of all ordinal
numbers α such that α < m. Thus for every ordinal number a,
(2)
a 6 Z ( m ) ο öc < m.
Let K (m) be the power of the set Z(m), so that
(3)
N ( m ) = l p .
We prove the following theorem:
4.1. For every cardinal number m, the power of the set of all ordinal
numbers α such that α < m is neither less than, nor equal to, the cardinal
number m, so that
(4)
~(K(m)
m).
The set Z(m), being a set of ordinal numbers, is well-ordered by the
relation "not greater t h a n " (see Theorem 3.2). Let an ordinal number
α be the order type of this set. Suppose that K(m) < m; it follows that
α = Z ( m ) < m. F r o m this and from (2) we infer that α e Z ( m ) . This
implies that Z ( a ) is an initial interval of Z(m), for if β e Z ( a ) , then
β < a and accordingly β
α ; m, so that ßeZ(m).
Hence Ζ(α)
c Z ( m ) . The type of Z ( a ) is the ordinal number a, but the order type
of Z(m) is also a, from which we infer that Z(m) is isomorphic to its
164
XI. WELL-ORDERED SETS
initial interval Z ( a ) , which differs from Z ( m ) (a e Z ( m ) , but α φ Z ( a ) ) .
This conclusion contradicts the theorem stating that n o well-ordered set
is isomorphic to any of its initial intervals other than itself (see the
proof of Theorem 2.1).
In particular, it follows from Theorem 4.1 that the set of all enumerable ordinal numbers, i.e., ordinal numbers α such that α < K 0 , is not
enumerable.
5. Theorem on transfinite induction. Transfinite sequences
The following theorem, called the theorem on transfinite induction,
is a generalization of the principle of induction for natural numbers
(see Chapter II, Section 1, Theorem 1.6).
5.1. Let (X, <) be any well-ordered set. If ψ{χ) is a propositional
tion which ranges over X and satisfies the following conditions:
func-
(1)
the first element of X satisfies the propositional function <p(x),
(2)
for every y e X, if every z e X, such that ζ < y and ζ Φ y,
the propositional function φ(χ), then y also satisfies φ(χ),
then every element of X satisfies the propositional function
satisfies
φ(χ).
T o prove the above assume that (1) and (2) both hold and that there
is in X an element which does not satisfy the propositional function
φ(χ). Thus the set Y of those elements y of X which d o not satisfy
the propositional function φ(χ) is a non-empty subset of X and
accordingly has a least element y0. It follows that, for every ζ eX, if
ζ < y ο and ζ Φ y0, then ζ satisfies φ(χ). Moreover, since y0 is not the
first element of X (in view of (1)), there is in X at least one element ζ
such that ζ < y0 and ζ Φ y0. It follows from this fact and from (2) that
y0 satisfies φ{χ), which contradicts the assumption made above.
Now let α be any ordinal number and let Z ( a ) denote the set of all
ordinal numbers less than a. Each function / : Ζ(α) -> X, where X is
any set, will be called a transfinite sequence of type α and, under the
convention adopted in Chapter III, Section 1, p. 37, will be denoted by
(Xß)ß**For every ordinal number β < α, f(ß) = xß is an element
5. THEOREM ON TRANSFINITE INDUCTION
165
of X. If, for any ordinal numbers β < α and γ < α, the condition
is called a one-to-one
β Φ γ implies the condition χβ Φ xy, then (χβ)β<Λ
sequence.
5.2. If there is a transfinite sequence (χβ)β<α
of all elements of a set
X, then this set can be well-ordered, that is, there is a relation which wellorders X. In particular, if that sequence is one-to-one, then there is a
relation which well-orders X into type a.
For every χ eX, let Ax be the set of all ordinal numbers β < α such
that χβ = χ. Since, by assumption, (χβ)β<α
is a sequence of all elements
of X, for every χ eX there is at least one ordinal number β < oc such
that χβ = χ and accordingly Ax Φ Ο. By Theorem 3.2, every set^4 x , for
χ eX, is well-ordered, and hence it has a least ordinal number. Let ξx
be the least ordinal number in Ax, and let Ζ = {Çx}xeX.
The set Z , being
a set of ordinal numbers, is well-ordered by the relation < . By setting,
for every χ eX, /(£x) = χ we define a one-to-one mapping of Ζ onto
X. A relation
is defined on X t h u s : for every x,y
eX,
(3)
χ
y if and only if ξχ <
ξγ.
It follows immediately from this definition that
is a reflexive, transitive, antisymmetric and connected relation, since all these properties
apply to the relation < in Z . Hence it follows that < * linearly orders
X. It follows from the definition of / and from (3) that / establishes an
isomorphism between ( Z , < ) and (X, < * ) , because it is a one-to-one
mapping and the condition: / ( f x )
if and only if
is
satisfied. We infer from this and from 1.5 that (X, < * ) is well-ordered.
is one-to-one, then Ζ = Ζ(α). Since
If the given sequence (χβ)β<Λ
( Ζ ( α ) , < ) is a well-ordered set of type α (Theorem 3.1), it follows
from the isomorphism between the sets ( Z ( a ) , < ) and (X, < * ) that X
is well-ordered by < * into type a. N o t e also that under that ordering
x0 is the first element in X.
5.3. If (Χ, Ο is a set which is well-ordered by < into type cc, then
there is a one-to-one sequence (xß)ß<0L of all elements of X.
Under the assumptions specified in this theorem, (X, < ) is isomorphic to the set Z ( a ) , well-ordered by the relation "not greater t h a n "
into type α (see Theorem 3.1). Hence there is a one-to-one function
166
XI.
WELL-ORDERED SETS
/ : Ζ(α) -> X which establishes this isomorphism. This function is the
required sequence.
Among the ordinal numbers we single out those which are in the
form β + 1 (see Section 2, p. 157). They are called isolated. But it is
not true that for every ordinal number α there is an ordinal number β
such that α = ß+\: an example is provided by the number ω. Those
ordinal numbers, other than 0, which are not isolated are called limit
numbers.
The following theorem is another formulation of the theorem on
transfinite induction.
5.4. Let {Xß)ß<(X be a one-to-one transfinite sequence of all elements of
a set X. If φ(χ) is a propositional function, ranging over X, which satisfies
the conditions:
(4)
x0 satisfies the propositional function
φ(χ),
(5)
for every isolated ordinal number ß = y+\9ß<oi9ifxy
φ(χ)9 then χβ satisfies φ(χ),
(6)
for every limit ordinal number β < OL9 if for every ordinal number
γ < ß9xy satisfies <p(x), then xß satisfies φ(χ),
then every term of the sequence (xß)ß<a,
the propositional function φ(χ).
satisfies
i.e., every element of X, satisfies
Assume that, for an ordinal number β < α, χβ does not satisfy φ(χ).
It follows from (4) that β Φ 0. The set of those ordinal numbers γ which
are less than α and for which xy does not satisfy φ(χ) is non-empty,
and does not contain 0. Let β0 be the least ordinal number in
that set. Then β0 Φ 0, and, for every ordinal number ξ < β0, χξ sata n
isfies φ(χ). If β0 is
isolated number, then it is of the form β0 = ξ -f- 1,
where ξ < f + 1 = β0 (see Section 2, example). Hence χξ satisfies φ(χ).
This and (5) imply that xßo satisfies φ(χ), which contradicts the assumption. If β0 is a limit number, then, since, for every ordinal number
ξ < βθ9 χξ satisfies φ(χ), hence it follows from (6) that xßo satisfies
φ(χ), which is impossible. The assumption that there is a β < α such
that Χβ does not satisfy φ(χ) results in a contradiction, and hence is
false. Thus Theorem 5.4 is proved.
6. THEOREM O N DEFINITION BY TRANSFINITE INDUCTION
167
6. The theorem on definition by transfinite induction
The following symbolism will be employed: for every set Z , and for
every ordinal number a, let Φ ( α , Ζ ) denote the set of all sequences / :
Ζ(β) -» Ζ for β < α (see Section 5, p p . 164-165). The empty sequence
is also assumed to be in Φ ( α , Ζ ) . The following theorem is called the
1
theorem on definition by transfinite
induction ):
6.1. For every set Z, every ordinal number oc, and every function h:
Φ(α, Ζ)
Ζ , there is exactly one transfinite sequence f defined for the
ordinal numbers β < α, such that
f(ß) = h(J\Z{ß))
(i)
for every β < α.
Before proving this theorem we explain its meaning. The theorem
states that if Ζ is any set and if α is any ordinal number, then, for every
function h which associates an element of Ζ with each sequence of
elements of Ζ of type β < α (the empty sequence included), there is
exactly one transfinite sequence / = ( z y ) y <a such that, for every ordinal
number β < α, the ß-th term f(ß) = zß is the value of the function
h at the sequence / restricted to Ζ (β), i.e., the sequence (ζγ)γ<β.
In
particular, we have
z0 = h(0)
Z\
=
(0 here denotes the empty sequence),
z
A(( o))>
2 2 = A((z 0 , Z , ) ) ,
We first show that there is at most one function / which satisfies
^
= A((Zy)y</l)>
condition (1). Assume that g also is a sequence, defined for all ordinal
numbers β < α, which satisfies condition (1), i.e., is such that
(2)
*) If we confine ourselves, in the formulation of Theorem 6.1, to the ordinal
number α = ω , then we can easily formulate a theorem on definition by finite induction.
XI. WELL-ORDERED SETS
168
Let φ(χ) be a propositional function ranging over the set of all ordinal
numbers β < α and defined thus:
(3)
β satisfies <p(x) if and only 'uf(ß) =
g(ß).
The ordinal number 0 satisfies φ(χ), a s / ( 0 ) = h(0) = g(0). Assume that
y satisfies φ(χ) for every ordinal number γ < β < α. This and (3) imply
that / ( y ) = g(y) for y < β, i.e.,/|Z(j8) = g|Z(j8). This and conditions
(1) and (2) yield f(ß) = g(ß); hence β satisfies the propositional function
φ(χ). On applying Theorem 5.1 on transfinite induction we infer that
every ordinal number β < α satisfies φ(χ), which, by (3), proves that
f=g.
Assume now that a 0 is the least ordinal number for which there is
n o function / that satisfies condition (1). In view of this and that part
of the theorem which has been proved above we infer that, for every
ordinal number β < α 0 , there is exactly one function fß which satisfies
the condition
(4)
ίβ(γ)
= ύ([β\Ζ(γ))
for every ordinal number y < β.
We infer that, for γ < β,
(5)
fß\Z(y
+
l)=f79
where Z(y + 1) is the set of all those ordinal numbers which are not
greater than γ. Hence
Λ(0=Λ(0
for ξ
<γ<β.
This yields the conclusion :
(6)
fß\Z(y)=fy\Z(y)
for γ < β.
A sequence / , defined on the set of all ordinal numbers β < α 0 and
having terms in Z , will be defined t h u s :
f(ß)=fß(ß)
forß<oi0,
/(«o) = Α((Λ(/0><«„)·
( 7 )
It will be proved that the sequence / so defined satisfies condition (1).
If γ < β < α 0 , then, by (7), (4), and (6), the following formula holds:
(8)
Λγ)
= f7(y)
= A(/ y |Z(y)) = A(/,|Z(y)) =
fß(y).
7. ZERMELO'S WELL-ORDERING THEOREM
169
Hence it follows that
(9)
f\Z(ß)=fß\Z(ß)
for
ß<*0.
N o w (9), (7) and (4) yield
(10)
f(ß) = fß(ß)
= h(fß\Z(ß))
= h(f\Z(ß))
for β < α 0 .
hence, by Definition ( 7 ) , / ( a 0 ) = A ( / | Z ( a 0 ) ) .
S i n c e / | Z ( a ) = (jß(ß))ß<«0,
This and (10) imply that the sequence / satisfies condition (1), that is,
that there is a sequence / , defined for all ordinal numbers β < α 0 and
having terms in Z , which satisfies (1), and this contradicts the assumption. The proof is thus complete.
Theorem 6.1 makes it possible to define transfinite sequences by induction. Examples of such definitions will be given in the following
sections (proofs of Theorems 7.2 and 8.1).
7. Zermelo's well-ordering theorem. Note on the axiom of choice
In the initial stage of its development set theory was based on intuition. Yet it turned out that the use of an intuitive concept of set,
without a precise formulation of its properties, results in paradoxes. F o r
instance, the assumption that there is a set of all sets or that there is
a set of all ordinal numbers (see Chapter VIII, Section 6, p . 119, a n d
Chapter XI, Theorem 3.4), which does not disagree with intuition, would
result in a contradiction. Thus it became necessary t o make set theory
more precise by tn ating it as an axiomatic theory *). The first system of axioms for set theory was formulated by E. Zermelo in 1904. At
present, various methods of axiomatizing set theory are known, but
further discussion would go beyond the scope of the present book. The
system of axioms given in Chapter I, Sections 6 and 9, is not complete,
but it covers those properties of the concept of set which are sufficient
1
for the arguments carrieds out in this b o o k (footnote ) , p . 22) and
usually also suffices in the applications of set theory in mathematics
(see p . 23 in Chapter I).
Among the axioms formulated by Zermelo, special mention should
l
) -ee Charter I. Section 10.
170
XT. WELL-ORDERED SETS
be made of the axiom of choice *) (see Chapter I, Section 9, Axiom
(VII)). At one time this axiom was objected to by some mathematicians because of certain of its consequences which seem paradoxical. In 1940, K. Gödel proved that no contradictions result if the
axiom of choice is added to those other axioms of set theory which
2
are accepted without r e s e r v a t i o n s ) . In 1963, P . J . Cohen proved
that the axiom of choice is independent of the remaining axioms of
3
set theory ) . In view of these results mathematicians commonly make
use of the axiom of choice, but usually, when doing so, they indicate
the fact that the said axiom is referred to in the proof of a given theorem.
Those proofs in which reference is made to the axiom of choice are
4
called ineffective ).
The proof of the following theorem, called the general principle of
choice, requires the application of the axiom of choice.
7.1. For every set X, there is a function ιυ {the choice function) which
associates with each non-empty subset of X exactly one of the elements
ofX.
Let X be any set. Consider the function / which is defined on the
family 91 of all non-empty subsets of X as follows:
(1)
f(Y)
= {Y}xY
for every Y e 91.
By Definition (1), the value of the function / for the argument Ye 91
(i.e., for every Y cz Χ, Υ Φ Ο) is the set of all ordered pairs (Y, y),
where yeY.
The family {f(Y))Y^x of all sets f(Y), for Ye 91 consists
of non-empty (as Υ Φ Ο) disjoint sets. For if Yx Φ Y2 and Yx, Y2 ξ 91,
then (Yi,yi) Φ (Y2,y2),
for every yx e Υλ and y2 e Y2. By the axiom
of choice (Chapter I, Section 9, Axiom (VII)), there is a set to which
1
) Formulated by E. Zermclo in Untersuchungen über die Axiome der Mengenlehre,
Mathematische Annalen 65 (1908), pp. 261-281.
2
) K. Gödel, The consistency of the axiom of choice and of the generalized
continuum hypothesis with the axioms of set theory, Annals of Mathematics Studies 3,
Princenton 1940.
3
) P. J. Cohen, The independence of the continuum hypothesis, Proceedings of
the National Academy of Sciences of the U.S.A. 50 (1963), pp. 1143-1148, 51 (1964),
pp. 105-110.
4
) Many proofs based on the axiom of choice were analysed by W. Sierpinski.
7. ZERMELO'S WELL-ORDERING THEOREM
171
contains exactly one element from each set f(Y) for Y e9\. Obviously,
tt) cz ÇRxX, and hence xo is a binary relation. Since, for every
Ye9Ï,
there is exactly one y e Y cz X such that (Y, y)ew,
i.e., such that
Ytvy, the relation n> is the required function.
The general principle of choice, and hence indirectly the axiom of
choice, will be used to prove the following well-ordering theorem of
Zermelo :
7.2. For every set there is a relation which well-orders
it.
Let X be any set. Assume that X = m, and that an ordinal number
α is the order type of the set Z ( m ) of all ordinal numbers of a power
not greater than m (see Section 4, formula (2)). By Theorem 7.1, there is
a function tt> which with every non-empty subset Y of the set X associates one element yeY. Assume that X cannot be well-ordered. Then for
where β ί ζ a, the set
X—{xy}y<ß
every transfinite sequence (Xy)y<p,
is non-empty, since otherwise X would be contained in {xy}y<ß
(see
Chapter I, Theorem 4.1, formula (6)), and would thus be a subset
of a well-ordered set (see Theorem 5.2). Hence a sequence
(xß)ß<a
of elements of X can be defined by transfinite induction as follows:
(2)
x0 =
(3)
xß = to(X-{xy}y<ß)
to(X),
for 0 < β < a.
is one-to-one, for if βχ < β < α, then xßi e
{xy}y<ß
The sequence (χβ)β^α
whereas xß = tt>(X—{xy}y<ß)
eX—{xy}y<ß
. Hence xßi φ xß. It follows
that the sets Z ( a ) and {.fy}^« are equipotent. Accordingly we obtain
(see Section 4):
(4)
N(m) = Z{m) = a = 1(7) = &
<
a
• .X
= m,
Œ
since {xß}ß<a
X> It follows immediately from (4) that K(m) < m,
which contradicts Theorem 4.1.
Theorems 7.2 and 2.5 yield the following trichotomy theorem for
cardinal numbers:
7.3. For any cardinal numbers m and η one of the following
tions holds:
(5)
m < n,
m = η,
η < m.
condi-
XI. WELL-ORDERED SETS
172
In fact, let X = m and Υ = η . By Theorem 7.2, the sets X and Y
can be well-ordered. By applying Theorem 2.5 we obtain either m < n,
or m = η , or η < m.
As a corollary to Theorem 7.3 and Theorem 2.2, Chapter VIII, we
have
7.4. Every cardinal number m which is the power of an infinite set
satisfies the inequality K 0 ^ m.
Indeed, let A = m, where A is an infinite set. Should Ä < K 0 , then
A would be equipotent with an infinite subset N0 of the set Jf of all
natural numbers. By Theorem 2.2, Chapter VIII, N0
is an
infinite
enumerable set, and hence it is equipotent with Jf. This would imply
r
A ~J ,
that is, A = X 0> which contradicts the assumption
that
Ä < K 0 · We conclude that A < K 0 cannot hold. This and Theorem 7.3
yield
K0 = A
or
K 0 < A,
i.e.,
K 0 ^ ™·
8. A proof of the Kuratowski-Zorn lemma
The Kuratowski-Zorn
lemma was formulated in Chapter IX (see
3.2). A proof of this lemma will be given in this section. Recall that
the lemma reads as follows:
8.1. Let (X,
) be an ordered set. If every chain A cz X has an upper
bound in X, then X has a maximal element. More precisely, for every
x0 e X there is a maximal element χ such that x0 < x.
Let (X,
) be an ordered set that satisfies the assumptions made
in the lemma. Assume that X = m. Suppose that for an element x0
of X there is in X no maximal element such that x0 < x. It follows
from this and from the definition of a maximal element (Chapter IX,
Section 2) that for every element y of X such that x0 < y there is an
element ζ eX such that y
ζ and y φ ζ.
We now use the general principle of choice (Theorem 7.1), and
hence, indirectly, the axiom of choice, to define by transfinite induction a sequence (ζβ}β<α,
where α is the order type of the set Z(m)
8. PROOF OF THE KURATOWSKI-ZORN LEMMA
173
of all ordinal numbers of power not greater than m. The inductive
definition is as follows:
z
0
(1)
- ^0)
(2)
for every isolated ordinal number β = γ + 1 , β < α, the term
ζβ e Χ is such that z y ^ ζβ and zy Φ ζβ,
(3)
for every limit ordinal number β < α, the term ζβ eX is an
upper bound of the set {zy}y<ß9
if that upper bound exists,
and zß = x0 otherwise.
We show that zy < ζξ (i.e., zy <: ζξ and zy Φ ζξ) for γ < ξ < α.
Let Ζ be the set of all ordinal numbers β < α, such that if γ < ξ
β,
then the above condition is satisfied. Obviously, 0 is in Z . Assume
that Ζ(β) cz Z . If β = ό + l, then we infer from (2) that
ζδ<ζβ.
Since δ < β, we have δ e Ζ(β), and accordingly δ 6 Ζ . It follows that
if γ < f < δ, then zy <ζξ. We infer from this and from ζδ -< ζβ that
β e Z . If β is a limit number and Z(/?) c z Z , then the set {zy}y<ß
is
a chain (see Chapter IX, Section 3). This is so because, for any ordinal
numbers y , ξ such that γ < β, ξ < β, one of the following conditions is satisfied: γ < ξ, γ = £, ξ < γ (see Theorem 2.4). Hence,
since ξ,γβΖ,
one of the conditions zy < ζξ, zy = ζξ, Z | ^< z y is satisfied.
It follows that z y < ζξ or z^
z y , which proves that { z y } y < /, is a chain.
It follows from (3) and from the assumption made in the theorem
that ζβ is an upper bound of this chain, so that z y , ζβ for every γ < β.
We infer from this and from the fact that Ζ(β) cz Ζ and β is a limit
number, that z y Φ ζβ for every γ < β, so that z y < ζβ for every γ < β.
Since the condition γ < ξ < β implies that z y -<z^, we infer that
β e Z. By Theorem 5.4 on transfinite induction we infer that every
ordinal number β < α is in Z . This proves that z y ^< Z | for γ < ξ < α.
Thus (ζβ)β<<χ is a one-to-one sequence. Hence the set { z ^ } ^ is equipotent with the set Z ( a ) . Accordingly,
(4)
(see Section 4, formula (1)). Also, since {zß}ß<(Xcz
(5)
ζ
{ β)ί><Λ
Χ =
m.
X, we have
Χϊ·
174
WELL-ORDERED SETS
We infer from ( 4 ) , ( 5 ) , the assumption concerning the ordinal number α
and formula ( 3 ) , Section 4 , that
K(tn) = Z(m) =
α < X = m,
so that, X(m)
m, which contradicts Theorem 4 . 1 . The proof of
Theorem 8.1 is thus complete.
9. The continuum hypothesis
Under the convention adopted in Section 4 , formula ( 2 ) , Z ( X 0
is the set of all ordinal numbers α such that oc ^ K 0 , that is, those
ordinal numbers which are order types of well-ordered enumerable
sets. By Theorem 3 . 2 , the set Z ( K 0 ) is well-ordered by the relation ^
("not greater than"). Let Ω denote the order type of the set Z ( K 0 ) .
Ω is an ordinal number. It will be proved that
9 . 1 . The set
Z ( K 0) =
Ζ(Ω),
where Ζ(Ω) is the set of all ordinal numbers α < Ω.
It follows from Theorem 3 . 3 that there is an ordinal number β
S
which is greater than any ordinal number in Z ( K 0 ) - Clearly Z(N 0 )
c Ζ(β). We will show that Z ( K 0 ) is an initial interval in Ζ(β). Assume
that γ G Z ( K 0 ) and that ξ < γ. Then ξ is the order type of a subset
of a well-ordered enumerable set of type γ, so that ξ e Z ( K 0 ) , which
proves that Z ( K 0 ) is an initial interval of Ζ(β). Hence (Theorem 1.3)
there is an ordinal number α < β such that Ζ(α) = Z ( K 0 ) - The set
Z ( a ) is of type a (Theorem 3 . 1 ) , and hence a = Ω. This and the preceding equation yield Ζ(Ω) = Z ( K 0 ) .
9 . 2 . The set Ζ{Ω) is non-enumerable.
Should it be enumerable, its order type Ω would be in Z ( K 0 ) = Z ( ß ) ,
which would imply that Ω < Ω, which contradicts Theorem 2 . 1 .
The power of the set Ζ(Ω) is denoted by K j . It follows from Theorem
9 . 2 that
(1)
175
EXERCISES
We now show that
9.3. The cardinal number X, is the number that immediately
follows
KoAssume that m < Κ ] , and let X = m. The set X is thus equipotent
with a subset Y of Z ( K 0 ) , different from Z ( K 0 ) . Assume that α is the
order type of Y. Since Y and Z(a) are isomorphic, m = Υ = Ζ ( α ) .
It follows that α < Ω, for should Ω
α, then Z ( ß ) cz Ζ ( α ) , and
hence K x = Ζ(Ω) ; Ζ(α) = m, which contradicts the assumption that
m < K j . Since α < Ω, Y is an enumerable set, so that m < K 0 , which
proves Theorem 9.3.
The problem as to whether N\ = c (see Chapter VIII, Section 5,
p. 116) has remained unsolved for many years, and the hypothesis
sating that X x = c is called the continuum hypothesis; 1940 saw the
appearance of a paper by K. Gödel in which he proved that the continuum hypothesis is consistent with the axioms of set theory *); in
1963 P . J . Cohen proved that the continuum hypothesis is independ2
ent of the axioms of set theory ) .
Exercises
1. Prove that no infinite subset of a well-ordered set is of type ω*.
2. Prove that the union of any family of initial intervals of a well-ordered set
(A, < ) is an initial interval of (A, <).
3. Let α and β be two ordinal numbers and let (A, < 0 and (B, < 2 ) be sets wellordered into types α and β, respectively, and such that AnB = Ο. The sum α + /? of
these ordinal numbers is defined to be the order type of the well-ordered set (A u 5 ,
< ) , where < is a binary relation in A^jB defined by the following: for any χ and y
which are in A u 5 ,
x < y o ((x e Α Λ y e B)v (χ e Α Λ y e Α Λ χ <ly)v(xeBAyeBAx
<2y)).
Prove that the relation < well-orders A^jB and that the sum <χ-\-β does not depend
on the well-ordered sets (A, < , ) and ( £ , < 2 ) , but only on their order types.
4. Show that
α + 0 = 0 + α,
2
ω+ 1 Φ 1+ω,
*) See footnote ) on p. 170.
3
) See footnote ) on p. 170.
2
(α + β) + γ = oc + tf + y ) .
XI. WELL-ORDERED SETS
176
5. What is the order type of any well-ordered set isomorphic to the set { 0 , 1 , 2, ...,
ω , ω + 1 , ω + 2 , ω + 3}, well-ordered by the relation < ?
6. Let α and β be two ordinal numbers and let (A, < A ) and (B, < 2 ) be sets
well-ordered into type α and β9 respectively. The product ccß of the ordinal numbers is
defined as the order type of the well-ordered set (AxB, < ) , where < is a binary
relation in Αχ Β defined by the folowing: for any elements (al9 bt) and (a2, b2) of
the product
AxB,
{flu bi) < (a2, b2)o((bi
< b2Abx
φ b2)v
(bx = b2Aal
<
a2)).
Prove that the relation < well-orders the set Αχ Β and that the product aß does
not depend on the choice of the well-ordered sets (A, < )t and (B9 < 2) , but only
on their order types.
7. Show that
α · 1 = 1 · α = α,
ω-2 Φ 2-ω,
(<χβ)γ = οί(βγ).
8. Show that
(α < £ ) = > ( y - f o c < +β)
γ
9
(α < β)=> (a + y <
β+γ).
9. Show that
(α < β) => (ya < γβ),
(α < β) => ( a y <
βγ).
10. Use the general principle of choice to prove that if A is an infinite set, then
Χυ < Λ.
CHAPTER XII
x
THE PROPOSITIONAL CALCULUS ) A N D ITS APPLICATIONS
IN MATHEMATICAL P R O O F S
1. Introductory remarks
Constructing proofs is a procedure which is characteristic of mathematics. Constructing proofs consists in obtaining certain theorems
from other theorems whose validity has been established earlier or
which have been accepted as initial theorems (axioms). Obtaining theorems from other theorems is based on what is called deductive
reasoning, which is an instrument of mathematics. Mathematical
2
logic ) , and especially two of its branches: the propositional calculus
and the functional calculus, is that discipline one of whose tasks
is to study the nature of the reasonings used in mathematics and to
establish criteria for their correctness. A n elementary discussion of
3
these two branches of mathematical logic ) , with special reference
to their applications in mathematical proofs, will be described in this
chapter and in Chapter XIII.
*) The origins of the propositional calculus go back to antiquity and are due to
the Stoic school of philosophy (3rd century B.C.), whose most eminent representative
was Chrysippus. But the real development of this calculus began only in the mid-19th
century and was initiated by the research done by the English mathematician G. Boole,
who is regarded as the founder of mathematical logic. The propositional calculus
was first formulated as a formal axiomatic system (see Section 13) by the eminent
German logician G. Frege in 1879.
2
) Special mention is due to the contributions of the Polish school of logic to
the development of this discipline.
3
) The propositional calculus and the functional calculus are not both introduced as formal systems in this book. It is only in Section 13 of this chapter that a
formal approach to the propositional calculus is presented.
178
ΧΙΓ.
PROPOSITIONAL CALCULUS AND ITS APPLICATIONS
2. Propositional connectives
Let χ,β,γ,...
be any statements made in mathematics. Such
mathematical statements will hereafter be called propositions. We are
intuitively inclined to ascribe to propositions one of the two logical
values *): truth or falsehood. Let 1 be the symbol of truth, and 0, that
of falsehood. F o r the propositions α, β, γ,
let w(a), w(ß),
...
denote, respectively, the logical values of those propositions. Under
this convention, w is the function which to every proposition assigns
its logical value. The expressions
(1)
w(a) - 1,
w(a) - 0
are read, respectively: α is a true proposition,
α is a false
proposition.
The following words or phrases: and; or; if..., then...; if and
only if; not; can be used to form new propositions or propositional
functions. They are called propositional connectives. In order to establish precisely the meaning of the propositions or propositional functions formed of propositions or propositional functions by means
of propositional connectives we have to establish the logical value
of the propositions α and β, α or β, if α then β, oc if and only if β,
not a, according to the logical values of the propositions α and β.
The symbol Λ (see Chapter 1, p . 5) will be used instead of the
word and, and will be called the symbol of conjunction or logical product. The proposition α Λ β will be called the conjunction or logical product of the propositions α and β, and α and β will be called factors
of that conjunction or logical product. An analogous convention is
adopted for propositional functions. In accordance with intuition,
a conjunction α Λ β is a true proposition if both of its factors are true
l
) There are systems of logic in which more than two logical values are assumed.
They are called many-valued logics, as distinct from two-valued classical logic. The
idea of many-valued logics we owe to the Polish logician J. Lukasiewicz (1920).
The American logician E. Post independently developed similar ideas at almost the
same time (1921). Other major systems of logic, other than the two-valued one,
are connected with the criticism of implication in classical logic (see Section 2, p. 184)
and with the philosophical interpretation of the foundations of mathematics which
is called intuitionism.
2. PROPOSITIONAL CONNECTIVES
179
propositions. If one of the factors, or both, are false propositions,
then the conjunction is a false proposition. This can be expressed in
the form of the following table:
w(a)
w((xAß)
1
1
0
0
(2)
1
0
1
0
1
0
0
0
The logical value of a conjunction, as it depends on the logical
values of its factors, can also be given in the form of the following
equations
l
(3)
1 A 1 = 1,
):
1 Λ 0 = 0,
0Λ1=0,
0 Λ 0 = 0.
<·
Let φ(χ) and ψ(χ) be propositional functions of one variable, x9
which ranges over a space X. A n element a eX satisfies the proposiiional function φ(χ)Αψ(χ)
if the proposition φ(α)Αψ(α) is true (see
Chapter I, Section 8). It follows from (3) that this proposition is true
tf and only if both φ(α) and ψ(α) are true, that is, if a satisfies φ(χ)
and at the same time a satisfies xp{x). Thus, for every a e l ,
(4)
a satisfies φ(χ)Αψ(χ)
if and only if a satisfies φ(χ)
and a satisfies
ψ(χ)·
We show that the following equation holds:
2.1. {xeX:
φ(χ)Αψ(χ)}
= {xeX:
φ(χ)} n{x e Χ:
ψ(χ)}.
An element a of X is in the set on the left-hand side of equation 2.1 if
and only if a satisfies φ(χ)Αψ(χ).
By (4), this condition is equivalent
to the condition that a satisfies φ(χ) and a satisfies ψ(χ), that is, the
condition that a is in both sets on the right-hand side of equation 2.1.
This last condition is satisfied if and only if a e {x eX:
i
<p(x)} n{x
eX:
) Notice that we are now defining an operation on the set {0, 1} of logical
values, which for simplicity we denote by the same symbol that we use for conjunction. Now (2) and (3) yield the equation w(a Aß) = w(cc) A w(ß), in which the symbol
Λ on the left-hand side is treated as the conjunction symbol, and on the right-hand
side as the symbol for the operation on the set {0, 1} defined by equations (3).
180
XII. PROPOSITIONAL CALCULUS AND ITS APPLICATIONS
ψ(χ)}. Equation 2.1 is thus proved. This lends substance to the fact
that conjunction is also called logical product, as it illustrates the close
relationship between conjunction and the intersection (product) of
sets.
Example
Consider the propositional function 2\x (x is divisible by 2) and
the propositional function 3\x (x is divisible by 3), where χ eJf (Jf is
the set of all natural numbers). The following equation results from 2.1 :
{xeJf:
2\XA2>\X]
= {xeJf\
2\x}n{xeJf:
3\x}.
This equation states that the set of all those natural numbers which are
divisible by both 2 and 3, i.e., the set of all those natural numbers
which are divisible by 6, equals the intersection of the set of all natural
numbers divisible by 2 and the set of all natural numbers divisible
by 3.
The symbol ν (see Chapter I, p . 5) will be used instead of the
word or, and will be called the symbol of disjunction or logical sum
The proposition a v j S will be called the disjunction or logical sum of
he propositions α and β, and α and β will be called summands of t h a t
disjunction (logical sum). A n analogous convention is adopted for
propositional functions. In everyday language the word or is used in
two different senses. In the first, a statement of the form α or β is accepted as true if at least one of the statements α and β is true; in the
other, the compound statement is accepted as true if one of the s t a t e
ments α and β is true, and the other is false. In mathematics, the word
or is used in its former sense. Hence one adopts the convention that
a disjunction cay β is true if at least one of the propositions α and β
is true. This convention is expressed in the following table:
(5)
w(a)
w(ß)
w(a ν β)
1
1
0
0
1
0
1
0
1
1
1
0
181
2. PROPOSITIONAL CONNECTIVES
The logical value of a disjunction, as its depends on the logical values
of its summands, can also be given in the form of the following equa1
tions ):
(6)
l v l = l,
lvO=l,
O v l = l,
OvO = 0.
Let φ(χ) and ψ(χ) be propositional functions of one variable, JC,
which ranges over a space X. A n element a of X satisfies the propositional function φ(χ)νψ(χ)
if and only if the proposition
φ(α)νψ(α)
is true (see Chapter I, Section 8). It follows from (5) that this proposition is true if and only if at least one of the propositions φ(α)9
ψ(α) is true, that is, if a satisfies at least one of the propositional functions φ(χ)9 ψ(χ). Thus, for every a e l ,
(7)
a satisfies φ(χ) ν ψ(χ) if and only if a satisfies φ(χ) or a satisfies
ψ(χ).
We show that the following equation holds:
2.2. {xeX:
<ρ(χ)νψ(χ)} = {xeX:
φ(χ)}ν{χ
eX\
ψ(χ)}.
An element α of
is in the set on t h e left-hand side of equation
2.2 if and only if a satisfies the propositional function
φ(χ)νψ(χ).
By (7), this condition is equivalent to the condition that χ satisfies
at least one of the propositional functions φ(χ), ψ(χ), i.e., the condition that χ is in at least one of the sets on the right-hand side of equation 2.2. This last condition is satisfied if and only if a is in the union
of those sets. This proves equation 2.2. It explains the fact that disjunction is also called logical sum, as it illustrates the close relationship
between disjunction and the union (sum) of sets.
Example
Consider the following two propositional functions: χ < 0 and
χ > 0, where x e M (@ is the set of all real numbers). By 2.2,
{x e 0t\ χ < 0 vx > 0} = { x e l : Χ < 0} u{x e 0t\ χ > 0 } .
The symbol ^ (see Chapter I, p . 1) will be written instead of
*) Equations (6) define an operation on the set {0, 1} of logical values, which
for simplicity we denote by the same symbol that we use for disjunction. Now (5)
and (6) yield the equation w(ccv β) = w(oc) ν Η>(/?) , in which the symbol ν on the
left-hand side is treated as the disjunction symbol, and on the right-hand side as
the symbol for the operation on the set {0, 1} defined by equations (6).
182
ΧΠ.
PROPOSITIONAL CALCULUS AND ITS APPLICATIONS
the word not and will be called the negation symbol The proposition
^ α is called the negation of the proposition a. A n analogous convention is adopted for propositional functions. In accordance with
intuition, the negation of a true proposition is a false proposition,
a n d the negation of a false proposition, a true proposition. This is
expressed in the form of the following table:
(8)
w(a)
H>(~ a)
1
0
0
1
l
o r by means of the equations )
(9)
~ 1 = 0,
~ 0 = 1.
Let φ(χ) be any propositional function of the variable χ ranging
over a space X. An element a eX satisfies the propositional function
^ <p(x) if and only if the proposition ^ φ(α) is true, i.e., if φ(α) is
a false proposition. This last condition is equivalent t o the condition
t h a t a does not satisfy <p(x). This yields the following conclusion: for
every a e l ,
(10)
a satisfies ~ φ(χ) if and only if a does not satisfy
φ(χ).
The following equation illustrates the relationship between negation and the complementation of sets:
2.3. {xeX:
~φ(χ)}
= -{xeX:
φ(χ)}.
An element a e X is in the set on the left-hand side of equation
2.3 if and only if a satisfies ~φ(χ)9
i.e., by (10), if a does not satisfy
<p(x). This last condition is equivalent to the condition that α φ {x eX:
<p(x)}, i.e., the condition that ae — {xeX:
φ(χ)}. This proves equation 2.3.
*) Equations (9) define an operation on the set { 0 , 1 } of logical values, which
for simplicity we denote by the same symbol that we use for negation. Now (8) and
(9) yield the equation w(~ a) = ~ H>(a), in which the symbol ~ on the left-hand
side is treated as the negation symbol, and on the right-hand side as the symbol
of the operation on the set { 0 , 1 } defined by equations (9).
2. PROPOSITIONAL CONNECTIVES
183
Example
Let Jf be the set of all natural numbers, and let p(x) be the p r o positional function inJf:
"x is a prime number". By 2.3 we have:
r
{xeJ :
~p(x)}
= -{xeJf:
p(x)}.
The meaning of the propositions in the form / / a, then β will now
be discussed. Such a proposition will be written in symbols t h u s :
α => β (see Chapter I, p . 2), and called an implication or a conditional with the antecedent α and the consequent β. The symbol => will
be called the implication symbol In everyday language a statement
a, then β is interpreted to mean that β can be inferred from a. This
interpretation differs from that given to it in mathematics. The following example will explain the meaning of the proposition / / a, then β
as understood in mathematics. Consider the following arithmetical
theorem: for every natural number n,
(11)
if6\n,
then 3\n.
Formula (11) is read: if η is divisible by 6, then η is divisible by 3. This
theorem is true for any natural number, hence, in particular, for 2, 3, 6.
Thus the following propositions are t r u e :
(12)
(13)
(14)
if 6|2, then 3|2,
"
if 6|3, then 3|3,
if 6|6, then 3|6.
It follows from (12) that an implication α => β in which both the antecedent α and the consequent β are false propositions is interpreted
as a true proposition. It follows from (13) that an implication α => β
with false antecedent α and true consequent β is interpreted as true.
Finally, it follows from (14) that an implication α => β in which both
the antecedent α and the consequent β are true propositions is interpreted as true. Thus one case remains to be examined, namely that
in which the antecedent of an implication is a true proposition, and the
consequent is a false proposition. An example is furnished by the implication
if 6| 12, then 6|5.
In accordance with intuition, this proposition is interpreted as false.
184
XII.
PROPOSITIONAL CALCULUS AND ITS APPLICATIONS
The analysis carried out above justifies adopting the following convention on the meaning of an implication α = > / ? . An implication
α => β is interpreted to be a false proposition if and only if its antecedent α is a true proposition and its consequent β is a false one. In
the remaining cases such an implication is interpreted as a true proposition. This convention may be expressed in the form of the following table:
w(a => β)
W(GC)
1
1
0
0
(15)
1
0
1
0
1
0
1
1
The logical value of an implication, as it depends on the logical
values of the antecedent and the consequent, can also be indicated
by the following equations *):
(16)
1 = > 1 = 1,
1 = > 0 = 0,
0 = > 1 = 1,
0=>0=1.
We now prove that the following equation holds:
2.4. For any propositions
α, β,
w(<x => ß) = w(~ α ν β).
The proposition ~ α ν β is false if and only if u>(~ α) = 0 and
w(ß) = 0 (see (5)). These conditions are satisfied only if w(a) = 1
and w(ß) = 0, that is, only if the proposition α => β is false. We have
hus established that νν(α => β) = 0 if and only if w(~ α ν β) = 0;
this means that the propositions α => β and ^ α ν β always have the
same logical value, which proves Theorem 2.4.
N o w let φ(χ) and ψ(χ) be any propositional functions of a variable
χ ranging over a space X. An element a eX satisfies the propositional
*) Equations (16) define an operation on the set {0, 1} of logical values, which
for simplicity we denote by the same symbol that we use for implication. Now (15)
and (16) yield the equation w(oc=>ß) = w(oc) => w(ß), in which the symbol => on
the left-hand side is treated as the implication symbol, and on the right-hand side
as that of the operation on the set { 0 , 1 } defined by equation (16).
185
2. PROPOSITIONAL CONNECTIVES
function φ(χ) => ψ(χ) if and only if the proposition φ{α) => ψ(ά) is true.
By Theorem 2.4, this holds if and only if the proposition ~ <p(a) ν ψ(α)
is true, that is, if and only if a does not satisfy φ(χ) or a satisfies ψ(χ)
(see (7) and (10)). This yields the conclusion t h a t : for every a e X,
(17)
a satisfies φ(χ) => ψ(χ) if and only if a does not satisfy φ(χ) or
a satisfies
ψ(χ).
We now prove the following equation:
2.5. {x eX: φ(χ) => y>(x)} = -{x
e l : φ(χ)} u{x e l :
y(x)}.
This is true because, by Theorems 2.4, 2.2, and 2.3, we have:
{x e Χ: φ(χ) => ψ(χ)} = {x e Χ: ~ φ(χ) ν ψ(χ)}
= {χ eX: ~ φ(χ)} ν{χ
eX:
ψ(χ)}
= -{χ
eX:
ψ(χ)}.
eX: φ(χ)}υ{χ
Example
Consider the following two propositional functions: 2\x and 3 | * for
χ eJf
(where Jf is the set of all natural numbers). Theorem 2.5 yields:
{ A : e ^ : 2 | j c = > 3 | x } = - {x ejf:
2\x} KJ{X eJf:
3|x}.
Thus the set of those natural numbers which satisfy the propositional
function 2\x => 3|x is the union of the set of all odd natural numbers
and the set of all natural numbers divisible by 3.
There is still one propositional connective left: if and only if We
shall use the symbol <^ (see Chapter I, p . 2) for if and only if
it is called the equivalence symbol. The proposition α ο β is called the
equivalence of the propositions α and ß. An equivalence <χο β is, in
accordance with intuition, interpreted as true if both propositions have
the same logical value, that is, are either both true or both false. This
convention is expressed in the following table:
w(a)
(18)
1
1
0
0
W(OL
1
0
1
0
1
0
0
1
β)
XII.
186
PROPOSITIONAL CALCULUS AND ITS APPLICATIONS
The logical values of an equivalence, as they depend on the logical
values of its component propositions, are also expressed by the following equations *):
(19)
=
l o 0
= 0,
0 o l = 0 ,
0 < ^ 0 = 1 .
The following equation holds:
2.6. w(a οβ)
= Η'((α => β) Α (β => α ) ) .
This is the case because w((a => β) Α (β => a)) = 1 if and only if
both w(a => β) = 1 and w(ß
α) = 1 (see (2)). It follows from (15)
that both conditions are satisfied if and only if either w(ot) = 1 and
w(ß) = 1 or w(oc) = 0 and w(ß) = 0, that is, if and only if w(oc ο β)
= 1. This proves equation 2.6.
Let φ(χ) and ψ(χ) be any propositional functions of a variable Λ'
ranging over a space X. An element a e X satisfies the propositional
function φ(χ) ο ψ(χ) if and only if the proposition φ(α) ο ψ(α) is true,
that is, by Theorem 2.6, if and only if the proposition (<p(a) => ψ(α)) A
Α(ψ(α) => φ(α)) is true. This condition is satisfied if and only if a satisfies the propositional function (φ(χ) => ψ(χ)) Λ (ψ(χ) => φ(χ)). This
yields the following conclusion: for every a e l :
(20)
a satisfies φ(χ) ο ψ(χ) if and only if a satisfies (φ(χ) => ψ(χ)) A
Α(ψ(χ) => φ(χ)).
Formula (20) and Theorems 2.1 and 2.5 immediately yield the following equation:
2.7. {x G Χ: φ(χ) ο ψ(χ)} = { — {x e Χ: φ(χ)} υ {χ e Χ: ψ(χ)})
η ( — {χ e Χ: ψ(χ)} u { x e Χ:
The propositional connectives:
=> (implication), and ο
Λ (conjunction),
ν
η
φ(χ)}).
(disjunction),
(equivalence), are binary propositional
connec-
tives, i.e., they enable us to form a new proposition or a new propositional function from two propositions or two propositional
x
func-
) Equations (19), similarly to equations (3), (6), (9), (16), define an operation on the set {0, 1} of logical values, which for simplicity we denote by the same
symbol that we use for equivalence. Now (18) and (19) yield the equation >v(a <=>/>)
— w(<x)ow(ß), in which the symbol <^> on the left-hand side is treated as the equivalence symbol, and on the right-hand side as the symbol of the operation on the
set {0, 1} defined by equations (19). The set {0, 1} together with the operations
Λ , V , ~ , =>, o , forms the two-element Boolean algebra.
2. PROPOSITIONAL CONNECTIVES
187
tions, respectively. The propositional connective ~ (negation) is a
unary propositional connective, i.e., it enables us to form a new p r o position or a new propositional function from one proposition or
one propositional function, respectively. All these propositional connectives share the property that the logical value of the propositions
formed by means of these connectives of certain given propositions
depends only on the logical value(s) of the given propositions, and
not on their meaning(s). Such connectives are called extensional. In
everyday language there are expressions which are propositional connectives but are not extensional. They do not play any role in mathematics and will not be discussed here.
The notation used in the present book is not the only one used in
mathematical or logical literature. Other symbols frequently employed
for propositional connectives are listed in the table below :
Negation
Disjunction
Conjunction
Implication
—a
Ntx
ä
~ a
a'
A(xß
OLVß
*vß
OL + ß
ccnß
KOL β
OL&ß
OL' β
*-ß
a^ß
Cocß
β
α ο β
OL* β
OL — >
Equivalence
Εχβ
χ~β
* = β
The first of these systems of notation is drawn mainly from t h e
algebra of sets and lattice theory. The second comes from the Polish
logician J. Lukasiewicz (note that here the binary connectives precede
the propositional variables and are not inserted between them; this
enables us to dispense with parentheses; Lukasiewicz's notation is also
known as the Polish notation and the parenthesis-free notation). The
third was used by D . Hilbert. The fourth comes from Peano a n d
Russell, while the fifth goes back to Schröder and Peirce.
The propositional connectives discussed in this section are not all the
extensional (unary or binary) propositional connectives.
An extensional unary connective o* enables us to form from any
proposition (or propositional function) a, a new proposition (or
propositional function) o*a, whose logical value w(o*oc) is defined
188
XII.
PROPOSmONAL CALCULUS AND I t S APPLICATIONS
in terms of w(a) by means of a table of type (8) or equations of type
(9). Thus there are as many unary connectives as many functions
2
/ : {0, 1} -» {0, 1} there are, that is, 2 = 4 .
The following table lists all the unary connectives:
w(o?a)
H-(a)
ο
1
0
!
0
1
0
0
1
w(o%ol)
1
1
It can easily be noted that o% is negation.
An extensional binary connective ο permits us to form, of any two
propositions (or propositional functions) α and /?, a new proposition
(or propositional function) α ο β, whose logical value νν(α ο β)
is defined from u>(a) and w(ß) by means of tables or equations of type
(2), (3), (5), (6), (8), (9), (15), (16), (18), (19). Thus there are as many
binary connectives as many functions / : {0,1} χ { 0 , 1 } -> { 0 , 1 } ,
4
there are, that is, 2 = 16.
All the binary connectives o f (/ = 1,
16) are listed in the table
on page 189.
The connective o2 is the conjunction, o8 is the equivalence, oi2 is the
disjunction, and Ö 1 4 is the implication.
It can be proved that all propositional connectives, whether unary
or binary, can be defined in terms of disjunction and negation. Note
also that there are binary connectives which suffice, each of them
separately, to define all connectives, whether unary or binary. These
connectives are o15 and o5. The connective o15 was produced in 1913
by Η. M. Sheffer, who called it alternative negation. The proposition
cco15ß is read: not both α and β. The negation of α can be defined as
α ο 1 5 α , and disjunction, α ν / ? , as ( α ο 1 5ο ί ) ο 1 5( β ο 1 5β ) . The connective
o s was termed by J. Lukasiewicz joint negation. The proposition cco5ß
is read: neither α nor β. Negation ^ α can be defined by o5 as α ο 5 α ,
and disjunction, α ν β, as (α o5ß)o5(ot o5ß). Note also that E. Zylinski
proved in 1925 that n o propositional connective other than o5 and oi5
suffices to define all the remaining connectives.
189
3. LAW IN THE PROPOSITIONAL CALCULUS
All the binary connectives are listed below:
w ( a ö A β)
1
1
0
0
o
1
0
1
0
1
0
1
0
w(a),
w(ß)
o0
1
1
0
0
1
0
1
0
1
0
1
0
H'(a o 3
β)
H>(a o 7
w (oc o 4
ft
0
0
1
0
β)
>v(a Os β)
1
1
0
0
1
0
1
0
1
0
0
1
H>(a o 1 0 /?)
W(OL oLlß)
w ( a σ 1 2/ ? )
0
1
0
0
1
1
1
1
1
0
1
w(aols
ο13β)
1
1
0
1
ft
0
1
0
0
H>(a Ö 6 β)
0
1
1
w(a
1
1
0
0
β)
0
0
0
1
w ( a o9
β)
1
0
0
0
0
H>(a os
1
1
0
0
w ( a o2
0
1
1
0
1
1
1
1
β)
W(CL Οί6
β)
1
1
1
1
3. The concept of law in the propositional calculus
Various schemata of compound propositions and propositional functions can be formed by means of propositional connectives. Examples
of such schemata are furnished by the formulas:
(1)
(2)
(oto
β)Λ
~
a,
( a V /8 ) = > ( ~ a
where a and β stand for any propositions or propositional functions
(as such they are propositional variables).
In correct inference an important role is played by those schemata of
compound propositions whose every instance is true regardless of the
XU.
190
PROPOSITION A L CALCULUS AND ITS APPLICATIONS
logical values of the constituent propositions. Such schemata are called
propositional laws or tautologies.
For a given schema A the notation \—A will be used to denote that
A is a propositional tautology.
Examples
(3)
|— Α => Α
(law of identity for
implication).
If w(y.) = 1, then, by formulas (15) and (16), Section 2,
W(A => y.) =
vv(a) => vv(A) = 1 => 1 = 1.
Likewise, if >v(a) = 0, then
w(x => y) = u'(a) => vi'(a) = 0 => 0 = 1.
(4)
|— Α ο A
(/aiv 0 / identity for
equivalence).
If w(y.) = 1, then, by formulas (18) and (19), Section 2,
w(a ο
Α) = W(A) <=> ΝΝ(Α) =
1 <=> 1 = 1.
Likewise, if w(oi) = 0, then
w(a) ο ΝΝ(Α) = 0 ο 0 = 1.
W(OÎ <=> Α) =
(5)
11— Α ν — Α
(/ΑΙΝ Ο/* excluded
middle).
If u>(a) = 1, then, by formulas (5), (6), (8), (9), Section 2,
w(oc ν — A) = w(oc) ν — w(A) = l v ^ l
=
lv0=l.j
Likewise, if w(a) = 0, then
VV(A ν — A) = VV(A) ν ^ w(A) = Ov ^ 0 = O v 1 = 1.
Law (5) states that of two contradictory propositions, i.e., propositions
one of which is the negation of the other, at least one is true. Application of this law is often made in proofs of mathematical theorems.
(6)
| - ^ ( Α Λ — Α)
(law of excluded
contradiction).
If w(oc) = 1, then, by formulas (2), (3), (8), (9), Section 2,
Η'(^(ΑΛ
—
A))
=
~(W(OL)A
~
w(A))
= ^ ( l Λ ~ 1) - ~ ( 1 Λ 0 ) = — 0 = 1.
Likewise, if w(a) = 0, then
ΙΝ(<^(Α Λ — A)) = — ( w ( A ) Λ — VR(A)) = ^ ( 0 Λ — 0)
= —( 0 Λ 1 ) = — 0 = 1 .
3. LAW IN THE PROPOSITIONAL CALCULUS
191
Law (6) states that of two contradictory propositions at least one is
false, in other words, two contradictory propositions cannot both be
true. Application of this law is also often made in proofs of mathematical theorems.
The method used above to determine whether schemata (3)-(6) are
propositional tautologies is general in nature and can be described as
a„ standing
follows. Let A be a schema formed from symbols a l 5
for any propositions, and from propositional connectives and parentheses. Schema A is a propositional tautology if and only if it stands
for a true proposition regardless of the logical values H ^ a J , Γ.., w(x„).
Each of the propositions α χ, . . . , ccn can be true or false. Hence we have
to compute, using the tables and equations given in Section 2, i.e. (2),
(3), (5), (6), (8), (9), (15), (16), (18), (19), the logical value of the proposition represented by schema A for all the possible combinations of the
a„. There are 2" such
logical values 0 and 1 of the propositions a l 5
cases. If in each case the logical value of the proposition represented by
schema A equals 1, then A is a propositional calculus tautology. If in
at least one case the logical value thus obtained equals 0, then A is not
a tautology, since it can be the schema of a false proposition. This
verification method is called the truth-table method. It makes it possible
to decide, in a finite number of steps, whether any given schema it
a propositional tautology or not.
The application of the truth-table method will be illustrated by
applying it to schemata (1) and (2).
Let w(a) = 1 and w(ß) = 1. Then
vr((a ο β) Λ ~ α) = (νν(α) ο w(ß)) Λ ^
w(a)
= ( 1 Ο 1 ) Λ - 1 = 1 Λ 0 - 0
by (18), (19), (8), (9), (2), (3), Section 2. Hence schema (1) is not a propositional calculus tautology.
We now show that schema (2) is such a tautology. Two symbols, α
and β, which occur in it, stand for any propositions. Thus we have to
compute the logical value of the proposition represented by schema
(2) for the following four possible combinations of the logical values
of the propositions α and β :
(a)
= 1, w(ß) = 1,
(b) w(a) = 1, w(ß) = 0,
192
XII.
PROPOSITIONAL CALCULUS AND ITS APPLICATIONS
(c) w(a) = 0, w(ß) = 1,
(d) w(a) = 0 , w{ß) = 0 .
Carrying out these computations, using formulas (5), (6), (8), (9),
(15), (16), Section 2, is left to the reader. On the other hand, it will be
shown here that (2) can be proved to be a propositional tautology in
a simpler way, by resorting to certain short cuts in reasoning. These
short cuts enable us t o leave unexamined some of the cases (a), (b),
(c), (d) above. The reasoning is as follows. If (2) is not a propositional
tautology, then there is a combination of the logical values of α and β
for which the proposition represented by schema (2) is false. The said
schema is an implication whose antecedent is (α ν β) and whose consequent is ( ~ α =>/?). By formula (15), Section 2, an implication is
false only if its antecedent is true and its consequent is false. In the case
under consideration this occurs when
(7)
But w(~
(8)
H'(avj8) = l
w ( ~ α =>/?) = 0.
and
α ν β) = 0 only when (see again formula (15), Section 2)
w ( ~ a) = 1
and
w(ß) = 0 .
Conditions (8) are satisfied only when (see formula (8), Section 2)
w(a) = 0
and
w(ß) = 0 .
But for these logical values of α and β we obtain
[w(a ν β) = w(a) ν w(ß) = 0 ν 0 = 0
(see formulas (5), (6), Section 2), which contradicts (7). This proves
that there is no combination of the logical values of α and β for which
the proposition represented by schema (2) could be false. This in turn
proves that (2) is a tautology.
The following theorem, which, given certain tautologies, describes
a method of constructing new tautologies, holds for all propositional
tautologies.
Let A be a schema consisting of symbols a 1 ?
a n of any propositions (i.e., propositional variables), of propositional connectives and of
parentheses. We signify this by the formula Α(αλ, ..., a„).
3.1. If a schema A(<xl9 ..., a„) is a propositional tautology, then on
substituting in A{cn1,
a„) schemata Al9
Anfor a l 5
a„, respectively, we obtain a schema which also is a propositional
tautology.
S . LAW IN THE PROPOSITIONAL CALCULUS
193
Let A* be a schema obtained from A(cc1, . . . , α Λ) by the simultaneous
substitution of Al9...9An
for <*i9...9ctn9
respectively. Let
be all those propositional variables which occur in Al9
determines w(Ax)9
Let wioii) = w(Ax)9
..., w(an) = w(An).
ßl9...9ßm
Any
associated with the
combination of the logical values w(ßx)w(ßm)
variables ßl9...9ßm
...9An.
w(An) and, in turn,
w(A*).
In this way, the prescribed com-
bination of the logical values of the propositional variables al9 . . . , α Λ
determines w(A)9 and does so in such a way that w(A) = w(A*). Since
A is a propositional tautology, we have w(A) = 1, and therefore w(A*)
= 1 regardless of the combination of the logical values associated with
the propositional variables which occur in A*. This proves that A* is
a tautology.
Example
On substituting in the law of excluded middle, α ν ^ α, the formula
(α Λ β) => γ for a we obtain a new tautology:
((α Λ β) => γ ν ) ^ ( ( α Aß) => γ).
N o w let 9>(χ), χ e Χ9 be any propositional function. We say that this
function is true in the set X if every element a οϊ X satisfies φ(χ)9 i.e., if
{x e Χ: φ(χ)} = X (see Chapter I, Section 8). We prove the following
theorem:
3.2. If a formula A is a propositional tautology, then by substituting
for the propositional variables in A any propositional functions of the
variables xl9 ..., xn that-range over the sets Xl9 ...9Xn9
respectively, we
obtain a propositional function of the variables xl9 ...9 xn which is true
in the set Xxx ... xX„.
Let A*(xi9
xn) be a propositional function obtained from a formula A by substituting propositional functions of the variables xl9 ...
for the propositional variables which occur in A. An element
...9xn
(al9
an) of the product Xx χ ... xXn satisfies A*(xl9
xn) if and
only if the proposition Α*(αί, ..., an) is true (see Chapter VI, Section 3).
But this proposition was obtained from the propositional tautology A
by substituting in it certain propositions for propositional variables, and
hence is true. It follows that every element of the product Xxx ... χ Xn
satisfies the propositional function A*(xt, ..., xn)9 which proves that this
function is true in the set Xt χ ... x l „ .
194
XII.
PROPOSITIONAL CALCULUS AND ITS APPLICATIONS
Example
Consider a propositional function obtained from tautology (2) by
substituting for α the propositional function χ < y 9 x9y e 0t9 and for
β the propositional function \z\ = 1, ζ e <€9 where 0t is the set of real
numbers, and # the set of complex numbers. The propositional function
in question is of the form
(x < yv
(5)
\z\ = 1) => ( ~ ( *
< j ) => | z | =
1).
By 3.1, this propositional function is true in the product
0tx
.
4. The concept of rules of inference. The rule of detachment
In mathematical proofs, all reasonings consist of very simple steps
which in turn consist in accepting certain propositions, or propositional
functions, as direct logical consequences of other such propositions or
functions. These elementary steps in deductive reasonings are called
rules of inference.
Let A1, ..., An be any finite sequence of schemata formed from propositional variables, propositional connectives, and possibly parentheses.
A schema Β will be said to be a logical consequence of the schemata
An9 and we write
Al9
Ai9 ..., An
(1)
Β
if the following condition is satisfied: for every combination of the
logical values associated with the propositional variables occurring in
Al9 ...9An9
B, if Al9 ...9An
stand for true propositions, then Β also
stands for a true proposition.
Rules of inference are operations which with finite sequences of
An associate a schema Β so that Β is a logical conschemata A19
sequence of Ai9 ...,An.
Rules of inference are written in the form of
(1), Ax, ..., An being called the premisses, and B, the conclusion *).
l
) The truth-table method is used to check whether a given formula in the form
of (1) is a rule of inference. To do so one has to determine whether the conclusion
is true for all those assignments of logical values to the propositional variables
which occur in the premisses Ai9
An and in the conclusion Β for which the premisses are true.
4. RULES OF INFERENCE. RULE OF DETACHMENT
195
An example of one of the most frequently used rules of inference is
the rule of detachment (modus ponens)
It can easily be verified that (2) is a rule of inference. Assume that
W(OL) = 1 and that w(oc => β) = 1. Should at the same time w(ß) = 0 ,
then, by formula (15), Section 2, and the assumption that w(oc) = 1, we
would have νν(α => β) = 0 , which would contradict the assumption that
w(a => β) = 1. Hence, w(ß) = 1, which proves that (2) is a rule of
inference.
It can easily be proved in a similar manner that, for any schemata
A, Β (formed from propositional variables, propositional connectives,
and parentheses), for every combination of the logical values associated
with the propositional variables occurring in A, Β such that w(A) = 1
and w(A
B) = 1 we have w(B) = 1. Hence it follows that also
(3)
is a rule of inference. This is a more general form of the rule of detachment.
Example
The use of the rule of detachment (2) in a mathematical proof will
now be illustrated. Let (an)nejr
be a given sequence of real numbers
which has been proved to be convergent. Thus the following proposition α is true :
The sequence (a„)ne^
(4)
is convergent.
At the same time, the following proposition α => β is also true:
(5)
The sequence (an)ne^r
bounded.
is convergent .=> the sequence
By applying rule (2) we accept the proposition β :
The sequence (an)„^r
(6)
i:is bounded
as true.
*) This rule was already known to the Stoics (3rd century B.C.).
(an)nej-
is
196
XII.
PROPOSITIONAL CALCULUS AND ITS APPLICATIONS
The rules of inference apply not only to propositions but also to
propositional functions. The following theorem holds:
4.1. Let ψι(χ), . · . , φη(χ), ψ(χ) be, respectively, propositional functions
obtained from the premisses Al9 ..., An and from the conclusion Β of the
rule of inference (I) by substituting propositional functions of a variable
χ which ranges over a set X for those propositional variables which occur
in these formulas. If an element a e X satisfies all the propositional funcfunction
tions ψι(χ),
φη(χ), then it also satisfies the propositional
ψ(χ). Consequently, if the propositional functions ψι(χ), ..., φ„(χ) are
true, then the propositional function ψ(χ) is also true.
Assume that the conditions specified in Theorem 4.1 are satisfied. If
aeX
satisfies the propositional functions φ^χ), ..., φη(χ), then the
propositions φ^α), ..., φη(ά) are true. These propositions and the proposition ψ(α) are obtained from the formulas Αί9 ..., An, Β by the substitution of certain sentences for the propositional variables which occur
in them. It follows from the definition of a rule of inference that if the
propositions ψί(α), ..., φη{α) are true, then the proposition ψ(α) is also
true. Thus, a satisfies the propositional function ψ(χ). If the propositional functions ψι(χ), ..., φη(χ) are true in X, then every element a e X
satisfies those propositional functions, whence it follows that every
element aeX satisfies ψ(χ). Thus ^(Λ;) is true in X.
Theorem 4.1 remains valid for propositional functions of many variables (see Chapter VI, Section 3), since they may be treated as propositional functions of one variable.
The rules of inference for the propositional calculus have the following property:
4.2. If all the premisses of a rule of inference are tautologies, then the
conclusion is a tautology, too.
This theorem follows from the definition of a propositional tautology
(see Section 3, pp. 189-190) and from the definition of rules of inference.
The theorem given below *) establishes the relationship between the
rules of inference and the propositional tautologies.
*) Theorem 4.3 is of special importance, since it makes it possible to obtain
rules of inference from propositional calculus tautologies, and conversely. For
instance, rule (2) yields the tautology (αΛ(α=>|5))=>|3. Hence, by Theorem 3.1,
5. EQUIVALENCE OF PROPOSITIONS
197
4.3. Let AY, . . . , AN9 Β be any schemata of propositions formed of propositional variables, propositional connectives, and parentheses. The schema
A
A
l9 >
(Ai A ... AAn)=>Bisa
propositional tautology if and only if
"" —»
Β
is a rule of inference.
Assume that (A^A ... AA„) => Β is a propositional tautology. Consider any combination of the logical values associated with the proposifor which every schema
tional variables that occur in AL9 ...9An9B
AL9 . . . 9 A N stands for a true proposition. We thus have w(AT) = ...
... = w(AN) = 1. This and formulas (2) and (3), Section 2, imply that
w(AiA ... AA„) = 1. Since, at the same time, ^ ( ( ^ Λ ... AA„) => B)
= 1, as (Αι A ... AA„) => Β is a tautology, hence, by formula (15) in
A
Section 2, w(B) = 1. This proves that
A
1
9
" ~ — " is a rule of inference.
Assume now that ( ^ Λ ... Λ A„) => Β is not a tautology. This implies
that for a certain combination of the logical values associated with the
propositional variables which occur in Αί9 . . . , AN9 Β the above schema
would stand for a false proposition. This is possible only if w(B) = 0
and w(AX A ... AA„) = 1. But a conjunction is true if and only if all
its factors are true (see formula (2), Section 2). This would imply both
w(Ai) = w(A2) = ... = w(An) = 1 and w(B) = 0 , which is not possible
ί 9
^ - is a rule of inference. Hence, if —L'-HL'-iiι j s a ri eu f 0
Β
Β
inference, then the schema under consideration is a tautology. The proof
of Theorem 4.3 is thus complete.
if ^
5. Equivalence of propositions and equivalence of propositional functions
Propositions α, β are called equivalent if they have the same logical
value. This is written as : α = β. The definition just given implies that
(1)
α = β if and only if W(OL) =
w(ß).
( A A ( A = > Bj) => Β is a tautology, where A and Β are any schemata of propositions consisting of propositional variables, propositional connectives, and parentheses.
On applying Theorem 4.3 we obtain rule (3).
198
XII.
PROPOSITIONAL CALCULUS AND ITS APPLICATIONS
Now, (1) and formula (18), Section 2, yield
(2)
α = β if and only if w(ct ο β) = 1.
Likewise, it will be said that the propositional functions φ(χ), x e Χ,
ψ(χ), x G Χ , are equivalent in Χ, which will be written a s : <p(x) = ψ(χ),
x G X, if for every a e X the propositions <p(a) and ψ(α) have the same
logical value. This condition is satisfied if and only if the propositional
true in X (see Section 3, p . 193), i.e., if
function φ(χ) ο ψ(χ), χ eX9is
every element a G X satisfies this propositional function. This definition
implies that
φ(χ) = ψ(χ), x G X, if and only ifw(q)(a))
(3)
=
w(y(a))
for every
and consequently,
(4)
for every a e X.
The definitions adopted above yield the following theorems:
5.1. For any propositions
α, β, γ, δ, the following
(5)
α = α,
(6)
if α = β, then β = α,
(7)
if α = β and β = γ, then α =
(8)
if α = β, then ~ α = ^
(9)
hold:
γ,
β,
if a = β and γ = δ, then oc Α γ = β Α δ, oc ν γ Ξ Ξ β ν δ, ce => γ
= β => δ, oc ο γ = β ο
δ.
5.2. For any propositional functions
the following relationships hold in X:
φ(χ) =
(10)
φ(χ),
ψ(χ),
σ(χ),
θ(χ),
xeX,
ψ(χ)9
if φ(χ) = ψ(χ), then ψ(χ) = <ρ(χ),
OD
if φ (χ) = ψ(χ) and ψ(χ) = σ(χ), then φ(χ) =
(12)
(13)
(14)
relationships
if φ(χ) = ψ(χ), then ~ φ(χ) =
if <p(x) = ψ(χ)
φ(χ)
ν σ(χ)
<=> σ(χ)
and
σ(χ)
=
ψ(χ)
ν θ(χ),
= ψ(χ)
ο
θ(χ).
=
σ(χ),
~ψ(χ),
θ(χ),
then φ(χ)Ασ(χ)
φ(χ)
=> σ(χ)
=
ψ(χ)
Ξ
Ψ(Χ)ΑΘ(Χ),
=> θ(χ),
φ(χ)
6. RULES OF DETACHMENT
199
The proofs of Theorems 5.1 and 5.2 follow immediately from (1)
and (3) and from formulas (2), (5), (8), (15), (18), Section 2.
If Γ is a fixed set of propositions (propositional functions in any
set X), then formula (1) (formula (3)) for α, β eT (φ(χ), ψ(χ) ΕΓ)
defines the relation = in that set. Theorem 5.1 (5.2) asserts that =
is an equivalence relation, which, moreover, satisfies conditions (8),
(9) ((13) and (14)).
The importance of Theorems 5.1 and 5.2 in deductive reasoning
should be emphasized. These theorems make it possible, at each step
of a deductive argument, to replace any proposition (propositional
function) by an equivalent proposition (propositional function).
Another important relation holding in any set of propositions (set
of propositional functions in any set X) will now be discussed.
It will be said that the proposition OL implies the proposition β , to
be written t h u s : α = > β , if w(a => β) = 1, i.e., if the proposition α => β
is true. Thus we have
(15)
α ξ ξ > β if and only if w(oc => β) = 1.
Let a relation < be introduced as an ordering relation on the set
{0, 1} of logical values; we assumed that
(16)
0<0,
1 < 1,
0 < 1,
0#
1.
On examining Table (15), Section 2, which establishes the logical values
of an implication in accordance with the logical values of the antecedent
and the consequent it is easy to see that
(17)
w(a => β) = 1 if and only if w(ot) <
w(ß).
Hence, (16) and (17) yield:
(18)
α = > β if and only if w(a) <
w(ß).
The following theorem holds:
5.3. For any propositions
(19)
(20)
(21)
(22)
(23)
α, β , γ, δ,
α=>α,
if α => β and β =} γ, then oc ξ ξ > γ,
if α ξ ξ > β and β = > α, then a = β ,
if a => β , then ~β
if oc => β and γ => δ, then χΛγ
= > — α,
=} β A Ô , oc ν γ =} β ν δ.
200
XII. PROPOSITIONAL CALCULUS AND ITS APPLICATIONS
By (16), w(a) < w(a), hence (19) holds ' ) .
It also follows from (16) that if w(a) < w(ß) and w(ß) ί ζ w(y),
then w(a) < w(y), which proves (20).
Likewise, using (16) we infer that if w(a) < w(ß) and w(ß) < w(a),
then u>(a) =
i.e., χ = β. This proves formula (21).
If oc =} β, then u>(a) ^w(ß).
If νν(α) = 0 a n d w(ß) = 1, then
w(/^ α) = 1 and w ( ^ j S ) = 0 , a n d hence
ß) -w(~
a ) . Likewise, if w(a) = w(ß), then w(—> /?) < w(— a ) . We have thus proved
that α = > β implies — β = > — α.
It now remains t o prove (23). T h e conditions α = > β a n d γ =} δ
are equivalent t o the conditions w(oc) ^ w(ß) a n d w(y) < w(<5), respectively. Should w(a v y ) < w(/8 ν ό) be false, then we would have
w(ß ν δ) = 0 a n d νν(α ν y) = 1. This, however, is impossible as
w(ß ν δ) = 0 implies w(/?) = vt>((5) = 0 , and consequently w(oc) = 0 and
= 0 ; but then w(a ν γ) = 0 . It follows that w(a ν γ) < w(ß ν δ).
w(y)
Likewise, should the condition w(oc Α γ) < u>(/? Λ δ) not be satisfied,
we would
have
w(aAy)
= 1 and
w(ßAÖ)
= 0.
Now
w(aAy) = 1
yields w(a) = w(y) = 1. Hence, as w(a) < w(/?) and w(y) < w(d), we
would have w(ß) = w(o) = 1, a n d hence w(ß Α δ) = 1, which contradicts w(ßAÖ) = 0. We have thus proved that w(oc Αγ) ^ w(ß Α δ),
and hence (23) holds.
As in the case of propositions, we shall say that a propositional
function φ(χ), x e l , implies a propositional function ψ(χ), x e X, t o be
written: φ(χ) => ψ(χ), x e Χ, if, for every a eX, \ν(φ(α) => ψ(α)) = 1.
Thus we have:
(24)
φ(χ) => ψ(χ), x e X, if and only if w(cp(a) => ψ(α)) = 1
for every a e X.
It follows from (24) and (17) that
(25)
φ(χ) => ψ(χ), x e X, if and only if w(<p(a)) <
w(y)(a))
for every a e X.
Theorem 5.3, (25) and (18) immediately give the following theorem:
*) Formula (19) is equivalent to the statement that Η- α => α.
6. RULES OF DETACHMENT
5.4. For any propositional
(26)
functions
φ(χ)
201
<p(x), ψ(χ), σ(χ), θ(χ),
=>φ(χ),
(27)
if φ(χ) =} ψ(χ) and ψ(χ) => σ(χ), then ψ{χ) => σ(χ),
(28)
ifφ(χ)
(30)
=} ψ(χ) and ψ(χ) => φ(χ), then φ(χ) = ψ (χ),
if φ(χ) =} φ(χ), then ~ ψ(χ) ΞΞ> ~
(29)
xeX:
//<ρ(χ) => ψ(χ) and σ(χ) =)θ(χ),
Φ
(Χ)9
then <ρ(χ) Λ CT(JC) = > ψ(χ) Λ Θ(Χ)
and φ(χ) ν σ(χ) =} ψ(χ) ν θ(χ).
Theorem 5.3 (5.4) states that in any set Γ of propositions (propositional functions in X) the implication relation = > is reflexive a n d
transitive, a n d hence is a quasi-ordering *). Moreover, it satisfies conditions (21)-(23) ((28)-(30)), t o be referred t o in t h e sections that
follow.
6. The rules of detachment for equivalence
For any propositions α, β, if a <=> β is a true proposition, i.e., if
w(ct ο β) = 1, then α = β a n d accordingly νν(α) = w(ß) (see Section 5, formulas (2), (1)). Hence, if a t t h e same time w(<x) = 1, then
w(ß) = 1. Likewise, if w(ß) = 1, then w(a) = 1. This yields the following two rules of inference:
1
ct9ocoß
β
}
β,
'
goß
a
They are called the rules of detachment for equivalence.
Example
The use of rules (1) in a mathematical proof will be shown below.
Let (an)nejr
be a fixed monotonie sequence of real numbers. By a wellknown theorem, valid in mathematical analysis, the following proposition holds for monotonie sequences:
the sequence (an)nG^
is bounded ο the sequence
(an)nejr
is convergent.
This may be represented by : α <=> β. N o w if it is proved that the sequence
]
) See Chanter IX. Section 5.
202
XII. PROPOSITIONAL CALCULUS AND ITS APPLICATIONS
(an)nejr
is bounded, then, by the first of the rules (1), we may conclude
that the sequence (a„)nejr
is convergent; and if it is proved that the seis convergent, then, under the second of the rules (2),
quence (an)nejr
we may conclude that the sequence (an)nejr
is bounded.
Rules (1) are very often used in mathematical proofs a n d regarded as self-evident.
7. Square of opposition
Mathematical theorems are usually in the form of a n implication.
If an implication α => β (where α and β are propositions or propositional function) is a theorem, then its antecedent α is called the hypothesis, and the consequent β, the thesis of the theorem. F o r instance,
in the following arithmetical theorem:
(1)
4\m => 2\m
(if m is divisible by 4, then m is divisible by 2), 4\m (m is divisible by 4)
is the hypothesis, and 2\m (m is divisible by 2) is the thesis.
If an implication α => β is a theorem, then α is also called a sufficient
condition for β t o hold, and β is called a necessary condition for α to
hold. F o r instance, in the case of Theorem (1), 4\m is a sufficient condition for 2\m t o hold, and 2\m is a necessary condition for 4\m t o hold.
For a given implication α => β, called the simple implication, the
implication β => α is called the converse implication. The truth of one
of them usually does n o t imply the truth of the other. F o r instance,
implication (1) is a propositional function which is true in the set of
all integers, whereas the implication converse to it is n o t an arithmetical
theorem as it is n o t true in the set of all integers.
Consider the following propositional calculus tautology, which is
called the law of contraposition:
(2)
|— (a => β) •»· (<~ β => ~ a).
The checking of the fact that (2) is a tautology is left t o the reader.
It follows from the law of contraposition that, for any propositions
(propositional functions) α, β,
(3)
(a => β) = (r^ β => ~ a ) ,
(4)
(β => α) ΞΞ ( ^ a => ~ β)
(see Section 5).
203
7. SQUARE OF OPPOSITION
The law of contraposition, taken together with the rules of detachment
for equivalence (see Section 6) a n d Theorems 5.1 a n d 5.2, makes it
possible t o replace, in any deductive argument, a proposition or p r o positional function in the form α => β by ^ /? => ^ α, and conversely.
F o r every simple implication α=>β,
the implication ~ β => ~ a.
will be called the opposite implication, while the implication ^ α => ^ β
will be called the contrary implication. Formulas (3) and (4) state that
a simple and an opposite implication are equivalent and that a converse
and a contrary implication are equivalent. These relationships are shown
in Fig. 18, which is called the square of opposition. Equivalent implicaα^β
β=>α
Fig. 18
tions are situated at the vertices of one and the same diagonal. It follows
from (3) and (4) that to prove all of the following implications: α => β,
β => α, ^ α => ^ β, ~ β => ^ α, it suffices to prove any pair of those
implications which are situated at one and the same side of the square,
since the remaining two implications are equivalent to those already
proved to be true. Each of the following pairs of implications: simple
and contrary, and converse and opposite, form what is called a closed
system of implications *).
Consider now the following propositional calculus tautology:
(5)
Κ(«ο/ϊ)ο((«=>/Ϊ)Λθ?=*α)).
*) The implications a x => ß t , ..., α π => ß n form a closed system of implications if
the antecedents cover all the cases so that ocj ν ... ν α „ is true, and the consequents
are pairwise exclusive. Then the truth of all converse implications: βγ => α ΐ 9 ..., βη
=> α„, results from the truth of those implications (K. F. Hauber (1775-1851)).
204
XII.
PROPOSITIONAL CALCULUS AND ITS APPLICATIONS
Formula (5) is a tautology by definition and by Theorem 2.6, Section 2. It follows from (5) (see Section 5) that, for any propositions
(propositional functions) α, ß9
α. ο β = (a=> β)Λ(β
(6)
=>oî).
Tautology (5), taken together with the rules of detachment for equivalence (see Section 6) and Theorems 5.1 and 5.2, enables us to replace, in any deductive argument, the proposition or propositional
function α ο β by the equivalent formula (α => β) Α (β => a ) .
Some mathematical theorems are in the form of an equivalence
αοβ.
This applies, for instance, to the following theorem: for any
sets X, Y,
χ = K o (Χ <= Υ)Α(Υ
Π)
ŒX).
Tt follows from (6) that in order to prove a theorem α ο β it suffices
to prove two implications : the simple one α => β and the converse
one β => α. Conversely, if α <=> β is a theorem, then the implications
α => β and β => ce are also theorems. In other words, β is then a necessary condition for a, and at the same time β is a sufficient condition
for a. Accordingly, a theorem of the form α ο β is often formulated
as: β is a necessary and sufficient condition for a.
For instance, Theorem (7) may be formulated t h u s : X cz Y and
Y cz X is a necessary and sufficient condition for X = Y.
We have already said that to prove a theorem of the form ceo β
it suffices to prove the simple implication α => β and the converse
one β =>oc. It follows from the square of opposition that to prove
α <=> β it suffices to prove one of the pairs of implications situated
in the square of opposition along one and the same side. Conversely ?
if α <=> β is a theorem, then all the implications: simple α => β, converse β => ce, contrary ^ α => ^ β, and opposite ~ β => ^ α, are
also theorems.
Examples
(I) Consider the following theorem in mathematical analysis:
(8)
If a sequence (a„)nej"
bounded.
is convergent, then the sequence (a„)nGj-
is
8. RULES OF HYPOTHETICAL SYLLOGISM
205
By the law of contraposition, Theorem (8) is equivalent to the following one (opposite implication):
(9)
If a sequence (an)nejr
not convergent.
is not bounded, then the sequence
(an)nej~
is
(II) Consider the following theorem in mathematical analysis: for
any set Χ φ Ο of real numbers
(10)
X has an upper bound if and only if sup X exists.
It follows from this (by means of the analyses carried out in this section) that the following implications are also theorems:
(11)
IfX has an upper bound, then s u p X exists.
(12)
If s u p Z exists, then X has an upper bound.
(13)
IfX has no upper bound, then s u p X does not exist.
(14)
If supX does not exist, then X has no upper bound.
We can infer from the square of opposition that to prove Theorem
(10) it suffices to prove any of the following pairs of theorems: (11),
(12); (12), (14); (14), (13); (13), (11).
8. The rules of hypothetical syllogism
The following rules of inference, called the rules of
syllogism *), are very often used in mathematical proofs:
hypothetical
α => β, β => γ
0)
y
(2)
(β => y) => (α => y)
9
ß=>y
(α => β) => (α => γ) '
To prove that (1) is a rule of inference assume that w(ct => β) = 1
and w(ß => γ) = 1. Should w(oc => γ) = 0, then we would have: W(OL)
= 1 and w(y) = 0 (see Section 2, formula (15)). This and w(a => β)
(3)
v
) The type of inference repiesented by the rules of hypothetical syllogism was
already known to the Stoics.
206
ΧΠ·
PROPOSITIONAL CALCULUS AND ITS APPLICATIONS
= 1 imply that w(ß) = 1, but we would then have w(ß => γ) = 1
0
x
= 0, which contradicts the assumption that w(ß => γ) = 1 ) . The
simple proofs of the facts that (2) and (3) are rules of inference are
left to the reader.
Example
An arithmetical theorem states that any number may be added
to both sides of an inequality. This may be written as a propositional
function
(4)
χ < y => x + z < y + z,
where
x,y,ze@t,
01 being the set of all real numbers. Another theorem states that both
sides of an inequality may be multiplied by any positive number. Hence
the following propositional function is true:
(5)
(x + z < y + z) => (t > 0 => (x + z) · t < (y+z)
· t),
where x, y, z, t e 0t.
Note that the consequent of implication (4) is the same as the antecedent of implication (5). By applying rule (1) to (4) and (5) we obtain,
as a result, the following theorem:
.*<>>=> ( i > 0 = > (x + z) - t < (y + z) - r ) ,
where x, y, z, t e
Examples illustrating the application of rules (2) and (3) are left
to the reader.
Some mathematical theorems assume the form:
Conditions <xx, ..., a„ are equivalent.
(6)
This means that
(7)
cci = ccj
fori
= 1,
j = 1,
It follows from Theorem 5.1 that to prove (7) it suffices to demonstrate
that
(8)
ocj = α 2 ,
α2 = α 3,
...,
α η_ χ
ΞΞ α„.
The following theorem enables us to simplify considerably the proof
of equivalences (8), and thereby the proof of theorems of the form (6)·
0 The proof of the fact that (1) is a rule of inference follows immediately from
Theorem 5.3, formula (20).
207
8. RULES OF HYPOTHETICAL SYXLOGISM
8.1. Let a l 5
a n be any propositions
for the equivalences (8)
αχ = a 2 ,
a2 = a 3,
to hold it suffices that the following
(9)
a x => a 2 ,
a2=>a3,
{propositional functions).
Then
α„_χ = a„
implications be true:
ccn^ => a„,
α„=>αχ.
We first show that
(10)
if implications (9) are true, then the implication αχ => α,· is true
for every y = 1, ..., w.
The proof is by induction. For y = 1 the implication α χ => α,· is true
as a => a is a propositional tautology (see Section 3, formula (3)).
Assume that the implication αχ => ccj is true for j = k, where Γ < f c
< n. It follows from the assumption made in Lemma (10) that the
implication ak => a f c +1 is also true. By applying rule (1) we infer that
the implication αχ => α Λ +1 is true, which completes the proof of (10).
We now show that
(11)
if implications (9) are true, then the implication α,· => a t is
true for every j = 1,
w.
F o r j = η the implication ccj => CLX is true under the assumption made
in Lemma (11). Assume that the implication is true for j = k, where
1 < / : < « . We infer from the assumption made in Lemma (11) that
the implication α Λ_! => cck is also true. By applying rule (1) to that
implication and to the implication <xk => a x we infer that the implication a f c_! => αχ is true, which proves (11).
N o w let i be any of the numbers 1, ...,n— 1. The implication
a, => a i + 1 is true under the assumption made in Theorem 8.1. T o prove
that off = di, !, i.e., that
ο oci, χ is true (see Section 5, formula (2))
it suffices to show that a i + 1 => a f is a true proposition (see Section 7,
formula (6)). It follows from (11) that the implication a i + 1 => αχ is
true, while the truth of implication αχ => a f follows from (10). On
applying rule (1) to these two implications we infer that the implication α ί +χ => οίι is also true. As ι is any of the numbers 1,
—1,
the equivalences a f = a i + 1 hold for any i = 1, . . . , « — 1 , provided that
implications (9) are true. The proof of 8.1 is thus complete.
208
ΧΠ·
PROPOSITIONAL CALCULUS AND ITS APPLICATIONS
Example
Consider the following theorem of the algebra of sets: for any sets
X, Y, the following conditions are equivalent:
(12)
XczY,
(13)
XuY=Y,
(14)
XnY
= X,
(15)
X-Y=0.
In order to prove this theorem it suffices to demonstrate that the implications:
x cz γ =>XuY = y,
XuY
= Y=>XnY
= X,
XnY
= X=> X-Y
= 0,
Ο => X cz 7 ,
X-Y=
are true.
9. Rules of inference involving conjunction and disjunction
The following rules of inference involving conjunction are very
often used in mathematical proofs:
(υ
a
-
4 ,
(XAp
(2)
(3)
(4)
«
χ
Α β
a
α βΑ
'
β
=> β,
α => γ
a => (β Λ γ)
α => β,
γ => ô
(oc Λ γ) => (β Λ δ) '
The proofs of the facts that (1) and (2) are rules of inferences follow
immediately from formula (2), Section 2, and from Section 4, for if
u'(a) = 1 and w(ß) = 1, then w(cc Α β) = 1, which establishes rule
(1), and if κ(α Aß) = 1, then νν(α) = 1 and w(ß) = 1, which establishes
rules (2).
9. RULES OF INFERENCE
209
Assume now that w(a => β) = 1 and w(ct => γ) -= 1. If so, then
νν(α => (β Α γ)) = 0 is impossible, for it would imply w(a) = 1
and w(ß Α γ) = 0 (see Section 2, formula (15)), and hence w(ß) = 0
or w(y) = 0 (see Section 2, formula (2)). But then we would have
w(<x => β) = 0 or u>(a => γ) = 0 , which would contradict the assumption. Hence w(a => (/? Λ γ)) = 1, which establishes rule (3).
In the case of rule (4) the proof follows from Theorem 5.3, formula
(23), and Section 5, formula (15).
Example
We show that, for any sets Χ, Y, Z:
if X a F a n d l c Z , then I c
YnZ.
The assumption X cz Y implies x e X => x e Y. Likewise, the assumption X cz Ζ implies x e X => x e Z. Both implications have the same
antecedent. On applying rule (3) we infer that
Z),
xeX=>(xeYAxe
which proves that X cz YnZ.
The above example illustrates the use or rule (3) in the proof of
a theorem in the algebra of sets. We suggest that the reader analyse
the proofs of the following theorems:
L i n i e X, for any sets X, Y;
2. If X cz Y and Ζ cz Γ, then XnZ
cz YnT,
for any sets X, 7, Ζ , Γ
In these proofs the reader will find illustrations of the applications
of rules (2) and (4).
Some of the most frequently used rules of inference involving disjunction are listed below:
(5)
ß
OLVß'
g
(6)
(7)
OCVß'
=» y> ß => y
a => β, γ => δ
(α ν γ) => (β ν δ)
*) Rules (2)-(7) are called algebro-logical.
1
210
ΧΙΓ. PROPOSITIONAL CALCULUS AND ITS APPLICATIONS
α ν /?, ~ β
(8)
ß
The proofs which show that (5) are rules of inference follow immediately from Section 2, formula (5), and the definition of rules of
inference as given in Section 4.
The proof which shows that (7) is a rule of inference can be carried
out just as in the case of (4) by using Theorem 5.3, formula (23), and
Section 5, formula (15).
T o prove that (6) is a rule of inference assume that w(oc => γ) = 1
and that w(ß => γ) = 1. Then νν((α ν β) => γ) = 0 is impossible, for
it would imply w(oc ν β) = 1 and w(y) = 0 (see Section 2, formula
(15)), so that we would have w(a) = 1 or w(ß) = 1 (see Section 2,
formula (5)), but then we would also have w(oc => γ) = 0 or w(ß => γ) = 0,
which would contradict the assumption. Hence, νν((α ν β) => y ) = 1
must hold, which proves that (6) is a rule of inference.
Assume now that νν(α ν β) = 1 and that w(~ a) = 1, so that vv(oc)
= 0. It follows from Section 2, formula (5), that w(ß) = 1 must hold.
This establishes the first of the two rules listed under (8). We proceed
to establish the other rule in a similar way.
Proofs of the following theorems in the algebra of sets offer simple
examples of applications of rules (5)-(7):
1. X a J u r , Y cz XuY, for any sets X, Y;
2. if X cz Ζ and Y cz Z , then XKJY CZ Z , for any sets Χ, Y, Z ;
3. if X cz Y and Ζ cz T, then XKJZ CZ YUT, for any sets Χ, Υ, Ζ, T.
Analysis of these proofs is left t o the reader. The above
theorems in the algebra of sets may be interpreted as analogues of
rules of inference (5)-(7), just as the theorems in the algebra of sets
quoted after the formulation of rules (2)-(4) may be interpreted as
analogues of those rules. Note that rules (2) and (5), Theorem 4.3, and
formula (15) in Section 5 imply that for any propositions α, β,
(9)
a Aß
= }
α,
α Aß
Ξ > β,
oc=}ocvß,
β =)αν
β.
Formulas (9) resemble formulas (7) and (8) in Chapter I, Section 3,
and formulas (7) and (8) in Chapter I, Section 2, valid for sets. Likewise, rules (3), (4), (6), (7) and formula (15) in Section 5 yield the following theorems: for any propositions α, β, γ, Ô,
211
10. SIMPLIFICATION RULES
(10)
if α = > β a n d α = > y , then α =}
(11)
if α ΞΞ> β and y ΞΞ> <5, then α Λ y = > / ? Λ ό
βΛγ,
(see Theorem 5.3,
formula (23)),
(12)
(13)
if α = > y and β = > y , then αν β =} y ,
if α Ξ=> /? and y ΞΞ> ό, then α ν y =} βν ô
(see Theorem 5.3,
formula (23)).
Similarity between formulas (10)—(13), on the one hand, and formulas
(9) and (10) in Chapter I, Section 3, and (9) and (10), Chapter I, Section 2, valid for sets, on the other, is easy to see. The latter are all those
theorems in the algebra of sets which are analogues of the rules of
inference under consideration *).
The first of the rules (8) is called modus tollendo ponens
a typical example illustrating its application.
2
) . Here is
Example
The law of trichotomy states that, for any real numbers χ and y, the
propositional function
(x < y ν χ = y) ν y < χ
is true. The propositional function χ < y ν χ = y is now replaced by
its equivalent χ < y. This yields the following true propositional function: χ < y ν y < x. On applying rule (8) to this propositional function
and to the propositional function ~(x < y) we infer that the premiss
^ ( J C < y) yields the conclusion y < x.
10. Simplification, Frege's, Duns Scotus' and Clavius' rules
The following rule of inference is called the simplification
rule:
β => oc
It states that if a proposition α is true, then every proposition β im1
) Close connections between rules (2)-(7) and theorems of the algebra of sets
justify calling these rules algebro-logical.
2
) The type of inference represented by rules (8) was already known to the Stoics.
212
XII.
PROPOSITIONAL CALCULUS AND ITS APPLICATIONS
plies a. The proof of rule (1) is very simple: it follows immediately from
formula (15), Section 2, that an implication whose consequent is true
is itself true regardless of the logical value of the antecedent.
l
The following two rules of inference are called Frege's rules ):
α:=> (β => y ) , α => β
α => (β => γ)
The first of them states that if α implies β => γ and if α implies β,
then α implies γ. The other states that if α implies β => y , then α => /?
implies α = > y . The easy proofs of these rules are left to the reader
as exercises.
2
The following rule of inference is called Duns Scotus' rule ):
(3)
It states that if ^ α is a true proposition, i.e., if α is a false proposition,
then α implies any proposition. The proof of rule (3) follows immediately from formula (15) in Section 2, since an implication whose antecedent is false is always true.
9
3
One more rule of inference to be given now is called Clavius rule ):
(4)
It states that if the negation of a proposition α implies a, then a is
a true proposition. The proof of rule (4) is very simple: should u ( ~ α
=> α) = 1 hold, then νν(α) = 0 would be impossible, since then we
would have w(~ α = > α ) = 1 = > 0 = 0. Hence w(a) must equal 1.
Rules (1), (2), (3), and in particular (1) and (3), are not frequently
1
) Rules (2) are analogues of the propositional tautology (α => (β => γ)) => ((a
=> β) => (a => γ)), which occurs as an axiom in the first axiom system of the propositional calculus, formulated by Frege in 1879.
2
) This rule is an analogue of the propositional calculus tautology ~ α => (ύ. => β),
which was already known to Duns Scotus, an eminent mediaeval philosopher who
lived at the turn of the 13th century.
3
) Reasonings based on rule (4) were already known to Euclid. Owing to Clavius,
a Euclid commentator who lived in the late 16th century, this type of inference became popular in scholarly circles.
213
11. APAGOGIC PROOFS
used in proofs of mathematical theorems. They have been mentioned
here because of their profound philosophical meaning and great importance in theoretical analyses.
Rule (4) finds application in what is called apagogic proofs, to be
discussed in the next section.
11. Apagogic proofs
Apagogic proofs are proofs by reductio ad absurdum. The method of
apagogic proofs consists in negating the theorem which is to be proved.
If the assumption that the theorem is false yields a contradiction, then
we conclude that the theorem is true. In this reasoning we use the following rule of inference:
(1)
~ α => ( / ? Λ
~
β)
It can easily be shown that (1) is a rule of inference, for if
α => (β Λ
^
/?)) =
1,
then
^ w(cc) => (w(ß) A ~ u>(/?)) = 1 ·
Since w(ß)A ~w(ß)
= 0 , ~ w ( a ) = 0 (see Section 2, Table (15)).
Consequently, w(a) = 1, which proves that (1) is a rule of inference.
Example
In the proof of Cantor's theorem (Chapter VIII, Theorem 6.5) use
was made of the following lemma: n o non-empty subset of a set X is
X
equipotent with 2 . The proof of that lemma was apagogic, and rule
(1) was used in it: the assumption was made that the lemma is false,
X
i.e., that there is a non-empty subset of X which is equipotent with 2 .
This assumption resulted in a contradiction, and accordingly we concluded that the lemma is true.
Rule (1) taken together with rule (3) in Section 3, and with the law
of identity γ => γ yields Clavius' rule (formula (4) in Section 10). In
view of the fact that ~ α => ^ α is a tautology we can, by rule (3),
Section 9, obtain from the implication ^ α => α the conclusion stating
XII.
214
PROPOSITIONAL CALCULUS AND ITS APPLICATIONS
that
α => ( α Λ ~ α ) . On applying rule (1) we obtain a. Thus a can
be inferred from ~ α = > α , which is immediately stated by Clavius'
rule. The proofs based on that rule are regarded as apagogic.
Example *)
The following arithmetical theorem will be proved: if a prime number
2
ρ divides a , then ρ divides a. Instead of: χ divides y we write: x\y.
The above theorem will now be written t h u s : for every prime number
ρ and for every integer a,
2
(2)
p\a =>p\a.
In the proof of this theorem we use the following arithmetical theorem:
for every prime number p, every integer a, and every integer b,
/>|β*=>(~(ρ|α) = > / # ) .
(3)
In particular, for every prime number ρ and every integer a,
(4)
p\aa => (~(p\a)
=> p\a).
It follows from Clavius' rule and Theorem 4.3 that
α => α) => α is
a propositional tautology (Clavius' law). Hence the propositional function
(5)
(~(p\a)
=>p\a) ^ p\a
is true. By applying the rule of hypothetical syllogism (formula (1)
in Section 8) we conclude from (4) and (5) that the propositional function (2) is true, which proves the theorem.
Rule (1) yields the following rule:
(6)
~ ( α => β) => (γ Λ
oc => β
~y)
It suffices to substitute in (1) (α => β) for a and γ for β (in this connection use is made of Theorems 4.3 and 3.1). Note t h a t for any propositions α, β,
~ ( a = > £ ) ΞΕ ( α Λ
(7)
x
~β),
) The theorem discussed in this example is a form of Euclid's theorem (Elements,
Book IX). The proof given here, which reflects the basic idea of Euclid's proof, is
usually quoted as an example of proof based, among other things, on Clavius' rule.
11. APAGOGIC PROOFS
215
for we have κ>(~(α => β)) = 1 if a n d only if νν(α => β) = 0 , which is
equivalent t o w ( a ) = 1 a n d w(ß) = 0 . Also, W(OLA ~ β) = 1 if a n d
only if w ( a ) = 1 a n d w(~ β) = 1, i.e., if a n d only if w(cx) = 1 a n d
(ß)
w
= 0 . Hence w ( ^ ( a = > / ? ) ) = 1 if a n d only if W(<XA ~ β) = 1,
which proves that
ν ν ( ^ ( α => /?)) = νν(αΛ ^ /?),
i.e., that equivalence (7) is true.
N o w (6) a n d (7) yield the following rule of inference:
(α Λ
~
β) => (γ Α
~γ)
Rule (8) is used if the theorem t o be proved by reductio ad absurdum
is in the form of an implication α => β. It is then assumed that α Λ ~ β
is true, a n d we try t o deduce a contradiction from this assumption. If
we succeed in doing so, then we infer that the implication α => β is true.
Often, when assuming α Λ ~ β, we arrive, by deductive reasoning, at
the conclusion ^ a . W e then use the following rule of inference:
(α Λ ~ β) => ~ a
(9)
•ß
which we obtain from (8), the tautology (α Λ ~ β) => ce, a n d rule (3),
Section 9. As (α Λ ~ β) ce => is a tautology, we can, by rule (3), Section
9, infer from ( α Λ ~ β) => ~ οι that (α Λ ^ β) = > ( A Λ ^ a ) . On applying rule (8) we obtain as a conclusion the implication α => β. Hence
α => β can be deduced from (α Λ ~ β) => ^ a , which is explicitly stated
by rule (9).
Example
Let m, n, k be variables ranging over the set./T of all natural numbers.
We shall write m\n for m is a divisor of n, a n d (m, n) = 1 for m, η
are prime to one another. T h e following theorem will be proved: for
all m , n, k e*/T,
(10)
~(m\k)
=> (/w|/fA: => (m, η) Φ 1 ) .
The proof will refer t o the following theorem: for every m, n, k eJf,
(11)
(m\nkA(m,n)
=
\)=>m\k.
216
XII.
PROPOSITIONAL CALCULUS AND ITS APPLICATIONS
Assume that m,n,k
are any natural numbers and that (10) does not
hold. By (7), this means that ~(m\k)
and ~(m\nk => (m9 ή) Φ 1).
By applying (7) once more we obtain ~(m\k), m\nk, and (m,n) = 1.
Since these conditions yield the antecedent of implication (11), using
the rule of hypothetical syllogism we infer that these conditions yield
m\k. We have thus obtained a result that contradicts the assumption
~(m\k). On applying rule (9) we obtain (10), which proves the theorem.
Sometimes, on assuming α Λ ^ β we arrive by deductive reasoning
at the conclusion β . The following rule of inference is then applied:
Π9Ϊ
(αΛ
~β)=>β
0C=>ß
·
By formula (2), Section 9, and Theorem 4.3, the schema (α Λ ~ β)
=> ~β
is a tautology. By rule (3), Section 9, (OCA ~β) => β and
(α Λ ^ β) => ~ β yield (α Λ ~ β) => (β Α ~ β ) . Hence, on applying
(6), we arrive at the conclusion α => β . Thus oc => β can be inferred
from (α Λ ^ β) => /?, which is explicitly stated by rule (10).
The proofs based on the application of the law of contraposition
(see Section 7, formula (2)) also are classed as apagogic. Instead of
proving a simple theorem oc => β we prove the opposite theorem ^ β
=> ^ α, which is equivalent to the simple one. The following two tautologies, also called laws of contraposition, are sometimes used as well:
(13)
(14)
\-(~oc=>ß)o(~ß=>oc),
| - ( a => ~ β) ο
(β => ~
a).
They are used, respectively, when the hypothesis or the thesis of the
theorem to be proved is in the form of a negation.
12. Principal propositional tautologies and their applications
Over a dozen tautologies in the propositional calculus which most
often find application in mathematical proofs will be listed below.
(1)
\-(oc=>ß)o(~ocvß),
(2)
h(avj8)o(~a=>j9),
(3)
((ocAß)=>Y)o(oc=>(ß=>
h
Y))
(law of exportation and
importation),
12. PRINCIPAL PROPOSITIONAL TAUTOLOGIES
(4)
L_
Λ β) Ο
(r*j
m
/
/
(6)
OL V ~
ß)\
mi
|— ~(OL => β) Ο (oc Λ ~ Β)
(7)
217
(De Morgan's laws) *),
(law of negation of implication),
j — ^ ( a Ο Β) Ο ( ^ ( a => β) ν ~(Β => a))
(law of negation of
equivalence),
(8)
|— ^ ^ a O oc
(9)
I— (α Λ a) ο
a,
(10)
\-(oL/\ß)<>
(Β AOL),
(law of double
negation),
(laws of idempotence),
\—(OLVOL)OOL
I - ( α ν β) Ο (Β ν OL)
(laws of com-
mutativity),
(11)
\-((*ΑΒ)ΛΓ)Ο(*Α(ΒΛΓ)),
\-((oLvß)vy)O(oLv(ßvy))
(laws of
(12)
(13)
I - ( α Λ (Β ν Γ))
associativity),
((OL Λ β) ν (α Λ y))
Ο
(/aw ο / distributivity
of conjunction over
1 - 0 ν (Β Ay)) ο
( ( a ν 0) A (OL V y))
(/tf w ö/ distributivity
of disjunction over
disjunction),
conjunction).
The proof of the fact that these formulas are propositional tautologies is left t o the reader. Laws (8)—(13) are called algebro-logical
Law (1) makes it possible t o define implication in terms of disjunction a n d negation. It states that, for any propositions or propositional
functions α and Β ,
(14)
0L=>ß=~0LVß
holds (see Section 5). This is the case because, by (1),
W((OL => Β) Ο
(~
OL ν Β))
=
1,
which, in view of formula (2), Section 5, proves that (14) holds.
Example
(I) The theorem in mathematical analysis which asserts that for every
sequence (an)ne%À
r,
if (an)neJ^ is convergent, then it is bounded, may, under
(14), be formulated t h u s : every sequence (an)nGjr,
is either not convergent or bounded.
1
) A. De Morgan (1806-1871), an English logician, discovered analogous laws for
the algebra of sets (Chapter I, Theorem 5.3). The laws (4) and (5) of the propositional calculus were formulated later, but they are usually called De Morgan's laws.
218
XU«
PROPOSITIONAL CALCULUS AND ITS APPLICATION S
Law (2) makes it possible to define disjunction in terms of implication and negation. It states that, for any propositions or propositional
functions α and β,
(15)
otvβ
ΞΞ ~ ot => β .
Example
(II) The law of trichotomy asserts that, for any real numbers, the
following propositional function is true:
X
=
JV(JC
<
j ν y
<x).
By (15), this propositional function is equivalent to the following one:
x^y=>(x<yvy<x).
Law (3) asserts that, for any propositions or propositional functions
α, β , γ, the formula
(16)
((ΧΑβ)=>γ
=
OL=>
(β=>γ).
Example
(III) Hypotheses in mathematical theorems are often in the
of conjunctions. For instance, in the arithmetical theorem
form
(x < y A z > 0) => (xz < yz)
the hypothesis is a conjunction of two propositional functions. By (16),
this theorem may be formulated in the following equivalent way:
χ < y => (z > 0 => xz <
yz).
In this formulation, χ < y is the hypothesis, and ζ > 0 => xz < yz is
the thesis.
It follows from De Morgan's laws (4) and (5) that, for any propositions or propositional functions α and β ,
(17)
~(<XAJ8) Ξ
α ν
~ β ) ,
(18)
-(αν^) = (-αΛ
~β).
(~
Equivalences (17) and (18) play an important role in proofs of mathematical theorems. They were used, for instance, in the algebra of sets
in passing from formula (1) to formula (2) in Section 3, Chapter I, and
in passing from formula (1) to formula (2) in Section 2, Chapter I. They
were also used in the proofs of De Morgan's laws for sets (see Chapter
12. PRINCIPAL PROPOSITIONAL TAUTOLOGIES
219
1, Theorems 4.2 and 5.3), although they were not stated explicitly. Examples of applications of equivalences (17) and (18) in various deductive
arguments will be given below.
Examples
(IV) D e Morgan's law, formula (13), Theorem 5.3 in Chapter I, will
be proved by means of (17):
xe
-(XnY)
= χφΧηΥ
= ~(xeXnY)
=
~(xeXAxeY)
= χ φ Xv χ φ Y = x e —Χ ν χ e — Υ = χ e —Xu — Y.
Hence —(XnY)=
— Xu — Y for any sets X, Y which are subsets of
any given space Z .
(V) In mathematical theorems, the hypothesis is usually a conjunction of two or more factors. F o r instance, one of the theorems in mathematical analysis states that, for every sequence (an\
(19)
((a„) is monotonie Λ (an) is bounded) => (an) is convergent.
The propositional function formulated under (19) is in the form (OCAβ)
=> y . The implication opposite to (19) is ~ γ => ~ ( α Λ / ? ) . By (17) and
Theorem 5.2 we obtain
~
γ => ^ ( α Λ β) Ξ ^
γ => (r^j
αν ~
β).
Hence the theorem under consideration may be formulated in the following equivalent form: for every sequence (a„),
(20)
(an) is not convergent => ((a M) is not monotonie ν
(an) is not
bounded).
(VI) In many mathematical theorems the thesis is in the form of
a conjunction. F o r instance, one of the theorems of arithmetic states
that if m divides the greatest common divisor of the numbers a and b,
then m divides a and m divides b. If we denote the greatest common
divisor of a and b by (a, b) we can write this theorem as follows: for
any natural numbers a, b,m,
(21)
m\(a,
b) =>
(m\aAm\b).
The propositional function (21) is of the form α=>(βΑγ).
The implica-
220
XÏÏ-
PROPOSITIONAL CALCULUS AND ITS APPLICATIONS
tion which is opposite to this one has the form ~(β
(17) and Theorem 5.2 we obtain
~(β
Λ γ)
γ) => ~ α. By
Λ
=> ~ α = (^ β ν ^ γ) => ^
α.
Hence the theorem opposite to (21) may be formulated: for any natural
numbers
a,b9m,
(22)
(~(m\a)
ν ~(m\b))
=> ~(m|(tf, b)).
(VII) The following definition is given in mathematical analysis:
(23)
a sequence (an) is monotonie ο
decreasing).
((α„) is increasing ν (a„) is
On applying (18) we infer that
(24)
a sequence (a„) is not monotonie ο ((«„) is not increasing Λ (an)
is not decreasing).
(VIII) Certain mathematical theorems have hypotheses in the form
of disjunctions. The following arithmetical theorem falls under this head:
for any integers a and b,
(25)
( α # 0 ν ^ 0 ) = > ( α , Α ) exists,
where (a, b) denotes the greatest common divisor of a and b. The propositional function (25) is in the form (ανβ)=>γ.
The opposite implication is ^ γ => ~ ( α ν β). By (18) and Theorem 5.2 we obtain
γ => ^ ( α ν β) Ξ Ξ ~
γ => (r^ α Λ ~
β).
Hence it follows that the theorem under consideration may be formulated in the following, equivalent, form : for any integers a and b,
(26)
(a, b) does not exist => (~(a
φ 0) Λ ~(b
φ
0)).
(IX) Some mathematical theorems have their theses in the form of
disjunctions. An example is provided by the following arithmetical
theorem: for every prime number ρ and for any integers a and b,
(27)
p\ab=>(p\avp\b).
The propositional function (27) is in the form a = > ( j 8 v y ) . The opposite
implication is ~ (β ν γ) => ~ α. By (18) and Theorem 5.2 we obtain
~ (βν γ) => ~ oc =
( ^
βΛ
^
γ) = > ^
ÖL .
221
12. PRINCIPAL PROPOSITIONAL TAUTOLOGIES
Hence the theorem opposite to the one under consideration assumes the
form: for every prime number ρ and for any integers a and b,
(28)
( ~ (p\a) A ~ (p\b)) => -
(p\ab).
The law of negation of implication (6) has often been used in the
present book. The equivalence
~ ( α = > β ) ΞΞ ( α Λ
(29)
~β),
where α and β are any propositions or propositional functions, follows
from (6); it was employed, for instance, in Section 11 in the example
on p . 215. It was used, in an intuitive manner, in the algebra of sets,
Chapter I, Section 1, when passing from formula (5) to formula (6).
Other examples of applications of equivalence (29) will be given in the
next chapter.
The following equivalence, which holds for any propositions or propositional functions α and β, results from law (7):
(30)
— (α ο β) = ( ~ (a => β) ν ~ (β => a ) ) .
Example
(X) It is known from the algebra of sets that any two sets X and Y
are equal if and only if the condition
(31)
xeXoxeY
is satisfied for every x. It follows from this that the sets X and Y are
not equal if there is an χ that does not satisfy condition (31); in other
words, that χ satisfies the propositional function ~ (x eX ο χ
eY).By
(30) and (29) we have
— (xeXoxeY)
= ( ~ (x e X => x e Υ) ν
~(xeY=>xeX))
=
φΧ)).
((x
GXAX
φΥ)ν
(χ
ΕΥΑΧ
Thus, Χ Φ Γ if and only if there is an χ that satisfies the propositional
function
(XGXAX
φΥ)ν(χε
ΥΑχφΧ).
The following equivalences, holding for any propositions or propositional functions oc, β, γ, result from (8)—(13):
(32)
^ α = α,
222
XII.
PROPOSITIONAL CALCULUS AND ITS APPLICATIONS
(33)
α Λ α Ξ α,
(34)
(35)
(Χ Α β
(οίΑβ)Αγ
(36)
=
ΞΞ
α ν α = α,
β ACL,
α Λ (β Λ y ) ,
αν β =
β ν oc,
(α ν β) ν y =
α Λ (β ν y) Ξ (α Λ β) ν (α Λ y ) ,
oc ν (β ν
γ),
α ν (β Λ y) = (α ν 0) Λ (α ν y ) .
We infer from (34) and (35) that in a conjunction (disjunction) of
more than two factors (summands) their order and grouping by
parentheses is irrelevant. This permits us to drop parentheses when
writing out a conjunction (disjunction) of more than two factors
(summands) and to rearrange their factors (summands) in an arbitrary manner.
The above equivalences have often been used in this book in proofs
of various theorems, even though reference to them was purely intuitive.
For instance, these equivalences were used in the proofs of identities (6),
Chapter I, Theorem 5.1; (6), Chapter I, Theorem 3.1; (6), Chapter I,
Theorem 2.1 ; (3), Chapter I, Theorem 3.1 ; (3), Chapter I, Theorem 2.1 ;
(4), Chapter I, Theorem 3.1; (4), Chapter I, Theorem 2 . 1 ; (14), (15),
Chapter I, Theorem 3.3, in the algebra of sets. We now show how those
identities can be proved by direct reference to equivalences (32)-(36).
Example
(XII) The identity (14), Chapter I, Theorem 3.3, will now be proved.
The definitions of the union and the intersection of sets and formulas
(36) yield
χ eXn(YvZ)
ν (χ GXAX
= x eX A (X e Υ ν x e Ζ) = (x e Χ ΑΧ e Υ) ν
eZ)
= xeXnYvxeXnZ
= χ G (ΧηΥ)
υ(ΙηΖ).
This proves that Ι η ( Γ υ Ζ ) = (XnY)v(XnZ)
for any sets Χ, Υ, Ζ .
Other applications of the propositional calculus in proofs of mathematical theorems will be given in Chapter XIII.
l
13. Axiomatic approach to the propositional calculus )
The propositional calculus, as described in the preceding sections
of this chapter, can be presented in a different way, that is axiol
) An axiomatic, formalized system of the propositional calculus will be described
in this section. The first formulation of the propositional calculus as a formalized
axiomatic system is due to G. Frege (1879K
1 3. AXIOMATIC APPROACH TO THE PROPOSITIONAL CALCULUS
223
matically. An axiomatic approach to the propositional calculus will
be given below.
Let Ρχ,Ρι,Ρτ,
— , Pi, ···> ieJi,
be all the propositional variables
that occur in the propositional calculus under consideration. Two
propositional connectives: negation and implication, denoted respectively by ~ and =>, are adopted. Propositional variables, linked by
propositional connectives, yield well-formed formulas. The set 3F of
all well-formed formulas *) is the least set that satisfies the following
conditions :
(f,)
for every / EJV', pt e J*~,
(f 2)
if A e^,
(f 3 )
then - ^ e # " ,
if A e & and Β Ε
then (A => B) e &'.
2
We make the following definitions ): for any well-formed formulas
A and B,
(AvB)
(i)
(2)
=
(AAB)
(AoB)
(3)
=
(~A=>B),
~ (A =>
= ((A => B)A
~B),
(B => A)).
The correctness of these definitions results from tautologies (2), (6)
and (8), Section 12, and tautology (5), Section 7, because these tautologies imply that the well-formed formulas on the left sides of (1), (2)
and (3) are respectively equivalent to the well-formed formulas on
3
the right-hand sides ) .
1
) The set of all well-formed formulas is the formalized language of the system
of the propositional calculus under consideration.
2
) These definitions can best be treated so that in each case the formula on the
left-hand side is a different way of writing down the formula on the right-hand side.
Hence, in every formula, formulas of the form of (~ A =>B), ~ (A=> ~ B) and
{A => Β) A (B => A) may be replaced, respectively, by the formulas (A V-B),
{AAB),
{Α ο Β), and conversely.
3
) In other words, for every assignment of logical values to the propositional
variables which occur in the formulas A and Β we obtain
w(AvB)=
w(~ A=>B),
w{A oB)=
W{AAB)
= W(~(A=>
w((A => Β) A (B => A)).
~
B)),
224
XII.
PROPOSITIONAL CALCULUS AND ITS APPLICATIONS
The following well-formed formulas are adopted as the axioms of
the propositional calculus:
(aj
(A => (B => A))
(a 2 )
((A =>(B=> C)) => (04
(law of
simplification),
B) => (A => C)))
(Frege's law),
( a 3)
( ~ A => (A => B))
(Duns Scotus' law),
( a 4)
( ( ~ A=>A)=>A)
(Clavius' law),
where A, B, C are any well-formed formulas.
It can easily be verified that axioms ( a j , ( a 2 ) , ( a 3 ) , ( a 4 ) are propositional tautologies.
The following rule is adopted as the rule of inference:
(4)
^ ' ^τΓ* ^
(modus ponens).
Β
Theorems of the propositional calculus are deduced from the axioms
listed above by application of modus ponens. If a well-formed formula
A is a theorem, we shall write |— A.
It is said that a sequence Al9 ...9A„ of well-formed formulas is
a formal proof of a well-formed formula A from any set ^ of wellformed formulas if the following conditions are satisfied:
(d,)
An = A,
(d2)
Αγ is an axiom or Ax e Ή,
( d 3)
for every 1 <j<n,
Aj is an axiom or Ai e ^ or Ai is obtained
from Λ ( and from (At => Aj) = Ak,
where 1
- / < j and 1
k
If for a well-formed formula A there is a formal proof from a set
of well-formed formulas, then this will be denoted by
^
< j , by the application of rule (4).
(5)
<3\-A.
In particular, if ^ = Ο, then we write
(6)
\-A,
13. AXIOMATIC APPROACH TO THE PROPOSITIONAL CALCULUS
225
which, in view of what has been said above, means that A is a theorem
of the propositional calculus.
Example
For every well-formed formula A,
(7)
\-(A=>A).
Let
Αι =
A2 =
(Α=>((Α*>Α)=>Α))9
(A=>(A=>A))9
A3 = ((A => ((A ^A)=>
A)) => ((A =>(A=> A)) => (A => Λ ) ) ) ,
A* = ((Λ =>(A=> A)) ^(A=>
A5 =
A)),
(A=>A).
The sequence of well-formed formulas Al9 A29 A3, A^, A5 is a formal
proof of the well-formed formula (A => A) from the set of well-formed
formulas ^ = O, for At and A2 are axioms in the form of (aO, A3 is
an axiom in the form of ( a 2 ) , ^ 4 is obtained from A3 = (A^ => Λ 4 ) and
from Ax by the application of rule (4), and A5 is obtained from A4.
t ne
= ( Λ 2 => ^ 5 ) by
application of rule (4).
As this example shows, the axiomatic method results in computations which are more difficult than in the case of the truth-table method.
The common object of both is t o determine the truth of propositions
or propositional functions obtained from well-formed formulas by
the substitution for propositional variables of given propositions or
propositional functions; in the case of both methods this goal is achieved
by performing a number of mechanical operations. Since the axioms
of the propositional calculus are tautologies, and rule (4) leads from
tautologies to a tautology (see Theorem 4.2), the theorems of the
propositional calculus are tautologies. Instead of using the truth-table
method in order to determine whether a given well-formed formula is
a tautology, in the axiomatic propositional calculus we search for
a formal proof of such a formula, which is a much more difficult task.
Axioms ( a j , ( a 2 ) , ( a 3 ) , ( a 4 ) have be selected so that every tautology
can be deduced from them by the modus ponens rule. It is possible
226
XII.
PROPOSITIONAL CALCULUS AND ITS APPLICATIONS
to choose other tautologies as axioms and t o adopt other rules of inference, but in any case the choice ought to be made in such a way
as to enable one to obtain all the tautologies of the propositional calculus from the axioms selected and by application of the prescribed
rules of inference.
Later on we shall prove a theorem which states that the axioms
adopted above, together with rule ( 4 ) , are sufficient to prove all the
tautologies. This theorem is called the completeness theorem. Before
proceeding to the proof we shall first define the concepts required
for this p u r p o s e and prove certain auxiliary theorems.
1 3 . 1 . DEDUCTION THEOREM. For every set & of well-formed formulas and for any well-formed formulas A and B9if/gv{A}\—B9
then
& I - (A =>B).
Suppose that the assumption made in the theorem is satisfied. Thus
there is a formal proof Al9
An of the well-formed formula Β from
the set &ν{Α}
of well-formed formulas. It suffices to show that, for
every / = 1 ,
& \— (A=>At).
The proof will be by induction.
If Ai = A9 then, by ( 7 ) , ^ |— (A => Ax). If A1 e & or if At is an axiom,
then, since (Ai => (A => Ax)) is the axiom ( a ^ , we may detach Ax
and thus obtain ^ |— (A => Ax). We have thus proved (see (d 2 )) that
Let 1
and assume that, for every 1 < y < k9 the condition & \- (A=> Aj) is satisfied. We show that
| - (A => Ak). If Ak
= A, or if Ak is an axiom, then the proof is similar to that for the case
Ax. Assume now that Ak has been obtained from the well-formed
formulas At and Aj = (Ai => Ak) (where i < k9 j < k) by application of rule ( 4 ) . By the inductive assumption, ^ |— (A => A{) and
<g \- (A=> Aj)9 hence ^ | - (A => (At => Ak)).
As the well-formed
formula
((A => (A, => Ak)) => ((A => A{) =>(A=>
Ak)))
is the axiom ( a 2 ) , by detaching (A => (At=> Ak)) first and (A => At)
next we obtain & [- (A => Ak).
A set 0 of well-formed formulas will be called consistent if the following condition is satisfied :
(8)
for every well-formed formula A, non ^ |— A or non ^ |— ~
A.
13. AXIOMATIC APPROACH TO THE PROPOSITIONAL CALCULUS
227
In other words, 0 is consistent if and only if <9\-A and & \- ~ A
d o not hold simultaneously for any well-formed formula A. It will be
proved that the consistency of # can be described by another condition, which seems rather strange.
13.2. For every set
tions are equivalent:
of well-formed formulas,
(9)
& is consistent,
(10)
the following
condi-
there is a well-formed formula A such that non & \— A.
Clearly condition (9) implies condition (10). T o prove that (10)
implies (9) we show that the negation of (9) implies the negation
of (10) (see Section 7). If ^ is inconsistenit, then there is a wellformed formula A such that 0 |— A and ^ |— ~ A. Let Β be any
well-formed formula. Since ( ~ A => (A => B)) is the axiom ( a 3 ) , by
applying rule (4) twice and by detaching from the above well-formed
formula ~ A first, and A next, we obtain a formal proof of Β from the
set 0 , so that ^ |— B. Thus condition (10) is not satisfied.
A set ^ of well-formed formulas will be called maximal if the following condition is satisfied:
(11)
for every well-formed formula A, & |— A or ^ |— ~
A.
We prove the following theorem:
13.3. For every set & the following
conditions are
equivalent:
(12)
& is maximal,
(13)
for every well-formed formula A, if non Ή \— A, then &v{A}
is
inconsistent.
Assume that condition (12) is satisfied. If non & \— A, then <3 \— ~ A
and, a fortiori, <&v{A} \- ~A.
Since &u{A} \- A, &v{A}
is inconsistent. N o w assume that condition (13) is satisfied. Let A be
any well-formed formula. If
\— ~ A, then ^ |— A or <3 \- ~ A
is satisfied. But if n o n ^ |— ~A,
then <&v{~A}
is inconsistent. By
13.2, <&KJ{~ A) \- A. This and 13.1 imply that & \- (~A=>
A).
Since ( ( ^ A => A) => Ä) is the axiom ( a 4 ) , by detaching ( ^ A => A)
we obtain # |— A, which proves that (13) implies (12).
228
XII.
PROPOSITIONAL CALCULUS AND ITS APPLICATIONS
13.4. For every set
of well-formed formulas and for every wellformed formula A, if & \— A, then there is a finite subset <&0 cz & such
that g0 I - A.
Let Al9 ..., An be a formal proof of a well-formed formula A from ^ .
e
Let <&0 ° the set of all those well-formed formulas Ax which are in ^ .
Then Αί9 . . . , Λ η is also a formal proof of the well-formed formula A
from & 0 .
The following theorem is a simple corollary of 13.4.
13.5. If a set & is inconsistent, then there is a finite subset <&0 cz &
which is inconsistent. It follows therefrom that if every finite subset of &
is consistent, then & is also consistent.
If V is inconsistent, then, for some well-formed formula A, & |— A
and <& \- ~ A both hold. By 13.4, there are finite subsets <§γ and ^ 2
of <3 such that ^ |— A and ^ 2 |— ~ A. The union <§χ u ^ 2 is a finite
subset of ^ such that <§x u ^ 2 | - A and ^ u ^ 2 h - ^ . Hence 9X u ^ 2
(
is a finite inconsistent subset of 3.
We now show that
13.6. Every consistent set
to a maximal consistent set
of well-formed formulas can be extended
of well-formed formulas.
The set ^ of all well-formed formulas is enumerable. Hence all
well-formed formulas can be arranged in an infinite sequence
(14)
A,,
A
such that every well-formed formula occurs in that sequence exactly once.
Assume that no consistent set <3' of well-formed formulas which
contains V is maximal. We can then define a sequence (&„) of sets
of well-formed formulas and a sequence (Bn) of well-formed formulas
thus : assume that
(15)
Since the set <SY is not maximal, it follows from 13.3 that there is a wellformed formula Β such that non ^t \— Β and such that 9T u{B} is
a consistent set. Let Bx be the first well-formed formula with this
property in sequence (14); we then define
(16)
V2
= Vi
Ufa}.
13. AXIOMATIC APPROACH TO THE PROPOSITIONAL CALCULUS
229
The set ^ 2 is consistent and contains ^ = 9T. Suppose that by continuing this procedure we have defined a set ^ „ and a well-formed
formula Bn_i9 where η > 2 . Since <3 α <§η and the set ^ „ is consistent,
it is not maximal. Accordingly, there is a well-formed formula Β
such that non ^ |— Β and such that yn\j{B}
is consistent. Let Bn be
the first well-formed formula with this property in sequence (14).
We then define
(17)
^ M +1 =
By definition, ^ „ c
and ^ n + 1 is consistent. Now define
^„u{2U.
00
(18)
<$* =
(J
We show that ^ * is a maximal consistent set of well-formed formulas
which contains 0 , which contradicts the assumption. In this way we
prove Theorem 13.6. It is worth noting that the apagogic proof a b o u t
to be produced is an example of the application of Clavius' rule (see
Section 10, formula (4), and Section 11). We want to prove that if
^ is a consistent set of well-formed formulas, then there is a consistent
and maximal extension of that set. We assume that ^ is consistent
and that there is no maximal consistent extension of ^ . F r o m this we
proceed to show that there is a consistent maximal extension of ^ .
In this way we establish the truth of a propositional function of the
type (Ο:Λ~β)
=> β, which is equivalent t o : α => ( ^ β => β) (see Section 12, formula (16)). Since under Clavius' rule ( ^ β => β) implies β,
we conclude that α implies /?, which is the essence of the theorem we
want to prove.
We now proceed to prove that ^ * is a maximal consistent extension
of ^ . N o w 0 cz ^ * follows immediately from the definition of
If ^ * is inconsistent, then by 13.5 a finite subset
of ^ * is inconsistent.
But
is contained in a set
whence it would follow that & η is
inconsistent, which is impossible as & n is consistent. If ^ * is not maximal,
there is a well-formed formula Β such that non ^ * |— Β and such that
0*u{2?} is consistent. Hence, for every neJf,
n o n ^ „ |— Β holds
and the set &nv{B}
is consistent. Since Β occurs in sequence (14)
and satisfies the above conditions, it would have to be one of the wellformed formulas Bj in the sequence (/?„), defined in the proof, and
230
XII.
PROPOSITIONAL CALCULUS AND ITS APPLICATIONS
accordingly we would have Β = Bj e VJ+l. Hence Β e
and accordingly V* I - B, contrary to the assumption. The proof of 13.6 is
thus complete.
13.7. If non [— α, then the set {~ a} is consistent.
If { ~ a} is inconsistent, then by 13.2 we have {~ a} [— a. This
and the deduction theorem (13.1) imply |— ( ~ α => α). This and axiom
( a 4 ) , on the application of the modus ponens rule, would in turn yield
|— a, which is contrary to the assumption of the theorem.
We now prove the completeness theorem. It states that
13.8. For every well-formed formula A, the following
equivalent:
(19)
(20)
conditions are
A is a tautology,
|— A (A is a theorem of the propositional
calculus).
Condition (20) implies condition (19) since, as we have already
said, axioms are tautologies and rule (4) leads from tautologies to
a tautology (see Theorem 4.2). It remains to prove that (19) implies
(20). Assume that non [— A (A is not a theorem of the propositional
calculus). By 13.7, the set {~ A) is consistent. This and Theorem 13.6
imply that there is a maximal consistent set 9 of well-formed formulas
that
such that {~ A) c
9 i.e., such
(21)
—
Ae<3.
Since ^ is a consistent and maximal set, it satisfies conditions (8) and
(11 ) ; in particular, for every propositional variable ρj, where 7 = 1,2, ...,
exactly one of the following conditions is satisfied: 9 |— pj9 9 |— ~Pj.
1
Let ν be the following valuation ) of all propositional variables, i.e.,
the function v: {pn}nejr
-> {0, 1}:
il
( 2 2)
!
^
H
o
if 9
\-pj9
if*,-~A.
) Every function which maps the set {pi}iejr of all propositional variables into
the set {0, 1} of logical values is called a valuation of the propositional variables.
Every valuation ν can be extended in a natural way so as to cover the set of all
well-formed formulas by means of the formulas: v(~ A) = ~ v(A),
v(A=>B)
= v(A) => v(B), and equations (9) and (16), Section 2.
13. AXIOMATIC APPROACH TO THE PROPOSITIONAL CALCULUS
231
We show that this valuation has the following property: for every
well-formed formula B,
if 9 I- B,
(1
*
( 2 3 )
( i
Ho
if
If Β is a propositional variable, then (23) holds in view of (22). Assume
that Β = ~ C, and that condition (23) is satisfied by C. If 9 \- B,
then 9 \- ~C and non 9 \— C, as the set 9 is consistent. Hence
v(C) = 0 , and accordingly
v(B) = v(~
C) = ~ v(C) = — 0 = 1 .
If non 9 I- B, then non 9 \- ~C.
If so, then 9 | - C , as the set 9
is maximal. Hence v(C) = 1, and accordingly
v(B)
=
v(~
C) =
~
Ü(C) =
—1=0.
Thus Β satisfies (23).
N o w assume that Β = (C =>£>), and that condition (23) is satisfied by the well-formed formulas C and D. Assume that 9 \— B, so
that 9 I- (C => D). If at the same time non & \- C, then v(C) = 0 ,
and accordingly
v(B) = v(C => D) = v(C) => *>(/)) = 0 => Ü ( £ ) = 1
(see Table (15), Section 2). If 9 \- C , then, since & \— (C => D), we
infer, by using modus ponens, that 9 |— Z>. If so, then ^(C) = #(Z)) = 1,
and accordingly
v(B) = v(C =>D) = v(C) => v(D) = 1 => 1 = 1.
Thus, if 9 h B> then v(B) = 1.
N o w assume that n o n 9 |— B, i.e., non ^ |— (C=> D ) . It follows
from this that non 9 |— D , for if 9 [— £>, then, as (Z) => (C => D)) is
axiom ( a j , on applying modus ponens we obtain 9 |— (C => D ) , which
is contrary to the assumption. Also we must have 9 |— C , for otherwise we would have 9 |— ~ C . Since the well-formed formula
C
=> (C => Z>)) is axiom ( a 3 ) , the application of modus ponens would give
us 9 \— (C => D), which is contrary to the assumption. Thus the condition non 9 I- (C => D) implies non 9 \— D and 9 \- C. Hence
9 \~ C and 9 \- ~D.
Consequently,
v(C) = 1
and
v(D) = 0.
XII. PROPOSITIONAL CALCULUS AND ITS APPLICATIONS
232
Hence
v(B) = v(C =>D) = v(C) => v(D) = 1 => 0 = 0 .
Thus, if non 9 | - B, then v(B) = 0 . T h e proof of the fact that the
valuation ν satisfies condition (23) is complete.
Since (21) holds for A, obviously ^ |— ~ A, a n d hence v(A)
= 0 *).
This proves that A is not a propositional tautology. We have thus
proved that if (20) is not satisfied, then (19) is not satisfied either, which
is equivalent t o the assertion that (19) implies (20). The p r o o f
of
Theorem 13.8 is thus complete.
Exercises
1. Define conjunction in terms of: (a) negation and disjunction, (b) negation and
implication.
2. Define equivalence in terms of conjunction, disjunction, and negation.
3. Determine which of the following schemata are propositional tautologies:
( ( a Af l ^ y ) = > ( a A ( ^ y ) ) ,
0L=>(ß=>(0LAß))9
((ce <=> β) ο )γ ο (β ο
(ce ο
γ)),
(α => ~ α) => ~ α.
4. Show that the following schemata are propositional tautologies:
(α=>βν(αΛ~0),
(((α => β) => α) => α)
(-/?=>-
(Peirce's law),
( α= > α ) ) = » 0 ,
( α ν β ) < = > ~ ( ~ α Λ ~ β).
5. Determine whether
~(α=>/0
α
'
~ ( Α= > # )
(<χνβ)=>γ
Λ => β9 ~ χ => ~ β
~ β
α => γ
αο β
OL => β, ~ α => β
α => β, OL => ~ β
are rules of inference.
J
) We have shown that there is a valuation ν such that v(A) = 0. The function
ν has been defined on the set of all propositional variables, but only finitely many such
variables occur in A, and v(A) depends on the values v(pî) only for i such that pi
occur in A. It follows that there is an assignment of logical values to the propositional variables occurring in A which shows that a corresponding instance of A is
a false proposition.
233
EXERCISES
6. Which rule of inference is the basis of the following argument, mentioned by
Origen (3rd c. A.D.), and probably known to the Stoics :
if you know that you are dead, then you are dead ;
if you know that you are dead, then you are not dead ;
hence you do not know that you are dead.
7. Prove that if a 1 } . . . , a „ cover all the possible cases, i.e., if the disjunction
α! ν ... v a„ is true, and if βί9 ..., βη are mutually exclusive, then the condition that
the implications
=> βί9 ..., α„=> βη are true implies that all the converse implications are also true.
The implications OLI=> ßiy ..., α„ => ft, form then a closed system of implications
(see note *), p. 203). Give an example of an application of the above, theorem
(formulated by Hauber) in a mathematical proof.
8. Formulate and establish a rule of inference under which the assumptions
\a\ >
2
2
2
\b\ =>a >
2
b,
(c > Ο Λ Α > b ) => ca
0A\a\
>
2
>
2
cb
yield
(c>
2
2
\b\)=>ca >
cb .
9. Carry out a logical analysis of proofs of theorems :
(a)
~ (a>
a),
by using the theorem : (a > b) => ~ {b > a),
(b)
(a > bA
~ (c = 0)) => - (ac =
be),
by using the theorems
(a > bAO
0) => ac>
be,
~ (a = b)o
(a>
bAc
(a > bva
< 0) => ac <
<
be,
b).
10. Prove, by means of the propositional calculus, the following equations, which
hold in the algebra of sets (see Section 12, Example (XII)):
AB
= Α —(Α η Β),
A =
(AnB)v(A-B),
An(B-C)=
(AnB)-C,
(A uB)~C
=
(A-Ou(B-C),
A-(B-C)
-
(A-B)v(AnC).
CHAPTER XIII
THE FUNCTIONAL CALCULUS A N D ITS APPLICATIONS
IN MATHEMATICAL PROOFS
2
1. Quantifiers ) and propositional functions of one variable
Next to propositional connectives (see Chapter XII, Section 2)
an important role in formulations of mathematical theorems and definitions is played by the words there is and every. For instance, it is
said that
(1)
(2)
for every natural number n9 η > 1,
there is a real number χ such that x
2
< 1.
If we denote the set of all natural numbers by Jf and the set of all real
numbers by 0tt we write
An>l,
instead of (1) and
V* <
2
1
xe&
instead of (2).
In general, if φ(χ) is a propositional function of the variable χ ranging over Χ Φ Ο (see Chapter I, Section 8), then instead of for every χ
in X, <p(x) holds we write
Ο)
Λ
ψ(χ),
xeX
*) The beginnings of the functional calculus (called also predicate calculus) can
be found in the 19th century, mainly in the works of C. S. Peirce (1839-1914),
E. Schroder (1841-1902), and G. Frege (1848-1925). Its modern form emerged
only in the early 20th century in the Principia mathematica (1910) by A. N. Whitehead and B. Russell.
2
) The term quantifier is due to the American logician C. S. Peirce.
1. QUANTIFIERS A N D PROPOSITIONAL FUNCTIONS OF O N E VARIABLE
235
and instead of there is an χ in X such that <p(x) holds we write
(4)
V<K*)xeX
This notation has been used in the present book from Chapter III
on (see Chapter III, Section 1) when various symbolic definitions were
made.
The symbols A
and V
xeX
a re
called, respectively, the symbol of the
xeX
universal quantifier which binds a variable χ ranging over X, and the
symbol of the existential quantifier which binds a variable χ ranging
1
over X ) . In writing out the quantifier symbols we often disregard the
symbol that indicates the range of the variable in question if that range
has been fixed previously and its omission does not result in any ambiguity. In such a case instead of (3) and (4) we write, respectively,
and
/\φ(χ)
X
"\/<p(x).
X
Attention must be drawn to the fact that the variable χ as it occurs
in formulas (3) and (4) has a quite different nature than has the variable
χ as it occurs in a propositional function <p(x). If, in a propositional
function φ(χ), the name of any element aeX is substituted for x, then
we obtain a proposition φ(α), which is either true or false. But if, in
formula (3) or (4), the name of any element of the set over which the
variable χ ranges is substituted for x, then we obtain a formula which
is meaningless. This is why the occurrences of the variable χ in (3)
and in (4) are called bound, i.e., bound by the quantifier in question,
whereas the occurrences of the variable χ in a propositional function
<p(x) are called free.
The distinction between free and bound occurrences of variables is
used not only in logic, but in mathematics as well. For instance, χ is a
2
free in x ,
1
3
but in J x dx χ is bound by the integral operator,
ο
*) The universal quantifier binding a variable χ is also denoted by the symbols
f^l, / 7 , (χ), V*.
X
The existential quantifier binding a variable χ is then denoted,
X
respectively, by U > Σ, (£χ),
Jx.
236
XtH.
FUNCTIONAL CALCULUS AND ITS APPLICATIONS
Quantifiers transform propositional functions of one variable into
true or false propositions.
The proposition
/ \ φ(χ)
is true if and only if every element a eX
xeX
satisfies the propositional function φ(χ), that is, if and only if the proposition φ(α) is true for every aeX.
By using the notation introduced by formula (5), Chapter I, Section 8, we can write the condition which is necessary and sufficient for
1
the proposition / \ ψ(χ) to be true as the following equivalence ):
xeX
(5)
Α
χ
s a
ψ( ) *
true
proposition
= {x eX: φ(χ)} = X.
xeX
The proposition
\J φ(χ) is true if and only if there is an element
aeX
xeX
which satisfies the propositional function φ(χ)9 that is such that the proposition φ(ά) is true.
The necessary and sufficient condition for the proposition \J φ(χ) to
be true can be formulated as the following equivalence:
(6)
\J
φ(χ) is a true proposition
xeX
Ξ {χ eX: φ(χ)} Φ Ο.
xeX
The following two equivalences result from (5) and (6), respectively:
(7)
/\
φ(χ) is a false proposition
= {x eX: φ(χ)} Φ Χ.
φ(χ) is a false proposition
= {xeX:
xeX
(8)
\/
φ(χ)} = Ο.
xeX
Examples
(I) Let A be any subset of a space X. Consider the propositional
function
χ e Ο => x e A,
χ eX.
The proposition a e Ο is false for every element a eX. Hence the implication a e Ο => a e A is a true proposition (see Chapter XII, Section
2, Table (15)). T h u s everyj'element aeX
satisfies the propositional
function in question, so that {x e X: x e Ο => x e A} — X. This proves
that the proposition f\ (x e Ο => x e A) is true.
xeX
l
) See Chapter XII, Section 5.
2.
QUANTIFIERS W I T H RESTRICTED RANGES
237
(II) Consider the following propositional function on the set 0t of
all real numbers:
2
χ Ε 0t.
χ -4χ + 3 = 0,
The element 3 of the set M satisfies this propositional function, and
2
hence { x e 0t \ x — 4x + 3 = 0 } φ Ο. We infer accordingly that the pro2
position V (χ - 4 x + 3 = 0 ) is true.
xe&
The universal quantifier may be interpreted as a generalization of
conjunction, for if the set X is finite, say X = {al9 ..., a„}9 then the
following equivalence holds:
(9)
A
<P(x) ο
α
(ψ( ι)
Λ ... Λ
ψ(αη)).
xeX
The proposition on the left-hand side of equivalence (9) is true if and
only if all the propositions ψ(α1), ..., ψ(βη) are t r u e , hence if and only i
the proposition on the right-hand side of equivalence (9) is true. This
proves that equivalence (9) is a true proposition.
Likewise, the existential quantifier may be interpreted as a generalization of disjunction, for if the set X is finite: X = {ax, ..., an}9 the following equivalence is true :
(10)
\/
φ(χ)ο(ψ(βί)ν
... ν ?>(*„)).
xeX
The proposition \J <p(x) is true if and only if at least one of the proxeX
position φ(α1)9 ..., φ(βη) is true, that is, if and only if the disjunction
φ(αί) ν ... ν φ(αη) is true. This proves the truth of ( 1 0 ) .
2 . Quantifiers with restricted ranges
Let φ(χ) and ψ(χ) be any propositional functions of a variable χ
ranging over Χ Φ Ο. In mathematics, the following locutions are often
used: φ(χ) holds for every x e X that satisfies the condition y>(x); there
is an Λ: Ε X, satisfying the condition ψ(χ) and such that φ(χ). F o r instance,
(1)
(2)
x
2
> 0 for every real number JC Φ 0 ,
2
there is a real number χ < 0 such that χ — 1 = 0 .
238
ΧΙΠ.
FUNCTIONAL CALCULUS AND ITS APPLICATIONS
Instead of ( 1 ) we write
Λ χ > ο,
2
(3)
and instead of (2) we write
(4)
V *
2
- 1 = 0 ,
having already specified that χ ranges over the set of real numbers.
In general, instead of for every χ that satisfies ψ(χ), φ(χ) holds we
write
(5)
Λ ?>(*)>
and instead of there is an χ that satisfies ψ(χ) and such that φ{χ)
we
write
V φ(χ),
(6)
y>(x)
having already specified the range of the variable x.
The quantifiers occurring in (3), (4), (5), (6) are called quantifiers
with range restricted by a propositional function. In (3), the propositional
function that restricts the range of the quantifier is χ Φ 0; in (4), it is
χ < 0; in (5) and (6), it is ψ(χ).
The following equivalences make it possible to pass from formulas
with quantifiers with ranges restricted by propositional functions to
formulas in which such quantifiers d o not occur:
(7)
ΑΨ(Χ)
=
ψ(χ)
(8)
Λ
O W
=>?(*)),
X
Vç>(*) = V ( * > ( * ) Λ ? ( * ) ) .
ψ(χ)
χ
T o prove ( 7 ) note that A <p(x) is a true proposition if and only if
ψ(χ)
every element a eX (where Xis the range of x) which satisfies ψ(χ) also
satisfies <p(x), i.e., if and only if
{x e X : y(x)}
cz {x eX:
φ(χ)}.
Under Theorem 5.5,
), Chapter I, formula ( 1 7 ) , this
thi condition is equivalent t o
•{xeX:
ψ(χ)} u{x e Χ: φ(χ)} = X.
3. QUANTIFIERS AND PROPOSITIONAL FUNCTIONS OF m VARIABLES
239
On applying Theorem 2.5, Section 2, Chapter XII, we infer that the
condition
e Χ: ψ(χ) => φ(χ)}
{x
= Χ
is equivalent to the previous one. But this condition holds if and only if
the proposition / \ (ψ(χ) => <p(x)) is true, which proves the truth of (7).
X
Note further that the proposition \J φ(χ) is true if and only if there
v(*)
is an element a eX that satisfies ψ(χ) and at the same time satisfies φ(χ),
which is equivalent to the condition that there is an element a e X which
satisfies the propositional function ψ(χ)Αφ(χ).
valent to the condition that \/(ψ(χ)
Λ <p(x))
This, in turn, is equiis a true proposition, which
χ
proves the truth of (8).
Using (7) and (8) we obtain, for instance,
2
/ \ χ
>
ο = Λ (* * ο => χ > ο)
2
χ^Ο
χ
V Χ - 1= 0
2
ΞΞ
χ<0
Χ/ (Χ < 0
Λ
Χ 2
1 = 0).
χ
3. Quantifiers and propositional functions of m variables
Consider any propositional function φ(χΐ9
"-,xm),Xi
eX± Φ Ο, ...
xm €Xm Φ Ο (see Chapter VI, Section 3). On prefixing this propositional function with the quantifier / \ or \ / (where j = 1 , . . . , m) we
..
_
Λ <
K
.
f*J
obtain a propositional function
(1)
*
i
o
r
x
^J
V
xfXj
X
9>(*i> ···>*»)
xfXj
of m — 1 variables.
Example
Consider the propositional function
(2)
xeY=>(x+l)eY,
χ eJf,
Ye2^,
where Jf is the set of all natural numbers, and 2^ is the family of all
240
XIII. FUNCTIONAL CALCULUS AND ITS APPLICATIONS
subsets of Jf. By prefixing propositional function (2) with the quantiwe obtain a propositional function of one variable Y:
fier
By prefixing propositional function (3) with the quantifier
we obtain
the proposition
(4)
When writing formulas constructed from propositional functions by
means of propositional connectives and quantifiers we parenthesize every
formula that we prefix with a quantifier. A parenthesized formula which
immediately follows a quantifier is the scope of the quantifier. A quantifier binds the occurrences of the variable concerned only if they are
within its scope. For instance, the quantifier
binds every
occurrence of the variable χ within its scope, but an occurrence of the
variable χ which is outside the scope of the quantifier in question is not
bound by it and remains free, provided it is not within the scope of
any other quantifier.
Example
In a propositional function of the form
the occurrences of the variable χ in φ(χ, y) are free since they are
not within the scope of any quantifier that binds x, whereas the occurrences of the variable χ in the formula ψ(χ, y) are bound since they are
within the scone of the Quantifie
The occurrences of the variable
y in the formula <p(x, y) and in the formula ψ(χ, y) are in both cases
bound by the quantifier
since they are within its scope.
In order to avoid an excessive accumulation of parentheses we adopt
the convention that quantifiers bind more strongly than do propositional
connectives, and that accordingly parentheses are always omitted when
there is no danger of any ambiguity in the interpretation of a given
formula involving quantifiers. In particular, parentheses will be omitted
3. QUANTIFIERS AND PROPOSITION AL FUNCTIONS OF m VARIABLES
241
if they indicate the scope of a quantifier t h a t refers to a simple formula,
such as φ(χ9 y, ζ ) , χ < y, x+y = ζ, etc.
Under this convention, formula ( 4 ) may be written as :
\J
Υ
f\{xeY=*x+leY),
χ
and the formula
V
X
<p(x)
=>
ψ(χ)
means the same as
V
(φ(χ))=>ψ(χ)·
X
We now give a few examples of applications of logical symbolism
in the recording of mathematical definitions a n d theorems.
Examples
(I) Cauchy's condition for the existence of a limit of a sequence (an)
of real numbers :
(an) is convergent «
where ε ranges over the set of real numbers, and the variables n0, k, η
range over the set of natural numbers.
(II) Cauchy's condition for the continuity of a function
at a point a e 0t\
f is continuous at
where the variables £, δ, χ range over the set 01 of all real numbers.
(Ill) T h e condition of linear independence of vectors a1, ..., an:
are linearly independent
where the variables at,
an range over the set of elements of a field
JT of real numbers, while α χ , . . . , an are in a linear space over
(IV) The principle of induction (see Chapter II, Section 1):
where Jf is the set of all natural numbers.
.
242
XUL
FUNCTIONAL CALCULUS AND ITS APPLICATIONS
Now let φ(χι,
x„), Xi G X ± ,
xn eXn,
be any propositional
function. An element (a2, ..., a„) e X2 x ... x ^ satisfies the proposiif and only if the propositioi
tional function
is true, that is, if and only if for every ax e X* the proposition
φ(αι,
a2,
aj
is true. The propositional function
true in the Cartesian product X2 x ... xXn if and only if it is satisfied by every element (a2,
of this product.
An element (a2,
an) eX2 χ ... xXn satisfies the propositional
,) if and only if the proposition
function
is true, i.e., if and only if there is an a1 eXx
ψ(αχ, a2,
(p(xL,a2,~.,an)
such that the proposition
..., a„) is true. The propositional function
(xi, ..., x ) is
n
true in the Cartesian product X2 χ ... χX n if and only if it is satisfied
by every element (a2, ..., an) of this product.
Examples
The propositional function
2
2
Λ ( ( * - > ' ) + Z = (x-y)(X+y)
zeä
+ z)9
X,VG
M,
is true in the Cartesian product 0t χ M because for every element (a, b)
G t% χ 0i the proposition
Λ
2
2
+ z = (a-b)(a
((a -b )
+ b) + z)
is true since the proposition
2
2
(a -b )
+ c = (a-b)(a
+ b) + c
is true for every c e M and for every (a, b) e 01 χ M.
The propositional function
\J x + z = y,
ze&
x, y G 01,
is true in the Cartesian product 0t χ 0t because for every element
(a, b) G 01 χ 0t the proposition \l
ze&
a + z = b is true.
243
4. FUNCTIONAL TAUTOLOGIES
The following theorem, which is a continuation of Theorem 5.2 in
Chapter XII, will now be proved.
3.1. For any propositional functions ψ(χι,
Xi Ε Χι,
... , Xn G Xn,
(5)
..., xn\
ψ(χί,
..., xn), where
if
φ(χί9
xn) = ψ(χΐ9
xn),
then
(6)
χ
χ
χ
Λ
<p(xi > · - > η ) = Λ ^ ( ι ' · · ' η )
Χι
Χι
and
Vv(*i> ···>*».) = V ^ n
(7)
*1
. . . , ^ Η ) .
-VI
Assume that (5) holds. Hence it follows that for every (al9
Ε Xn x ... x Xn the propositions φ(α1,
..., an) and χρ(αγ, ..., an) have the
same logical value. Hence, for every (a2, ..., a„) eX2
propositions Λ «ρΟ*!, a , . · ·, #„) and / \
χι
..·,#„)
2
x ... x A ^ , the
, α , . . . , α„) also have the
2
Xl
χ
a
same logical value. This is so because the proposition / \ ψ( ι>
i>
···
.... an) is true if and only if for every ax e Xl the proposition φ(αι,
...,
an)
is true, which is equivalent t o the condition that the proposition ip(ax,
...
..., an) is true for every ax eXl9
position
that is, to the condition that the p r o -
Α ψ(Χι, ci2, .··, an) is true. We have thus proved (6). The
proof of (7) is analogous.
4. Functional tautologies
Chapter XII was concerned with the laws of the propositional calculus, that is, with schemata of true propositions and propositional
functions, constructed from propositions or propositional functions
by means of propositional connectives and parentheses, so that their
truth results simply from the way in which they are constructed.
New propositional functions and propositions can be constructed
of given propositional functions when quantifiers are used in addition to propositional connectives. In mathematical proofs, an imp o r t a n t role is played by schemata of true propositional functions
XIII. FUNCTIONAL CALCULUS AND ITS APPLICATIONS
244
x
and propositions constructed of initial propositional functions ) by
means of propositional connectives and quantifiers in such a way that
their truth results simply from their structure, that is, the way they
are constructed, and not from the meaning of those non-logical concepts which occur in them. Such schemata will be called functional
laws or tautologies. \—A will be written instead of "A is a functional
tautology". It follows from Theorem 3.2, Chapter XII, that
4.1. If A is a propositional tautology, then by substituting for propositional variables in A any propositional functions or propositions
constructed from initial propositional functions by means of propositional
connectives and quantifiers we obtain a true propositional function or
a true proposition (i.e., true in an appropriate set).
Theorem 4.1 provides for a method of constructing
tautologies.
functional
Example
Consider the following propositional tautology:
h ( a = > ^ ( ^ a v β)
(see Chapter XII, Section 12, formula (1)). If in this tautology Λ φ(χ, y)
χ
is substituted for α and \ / \p(x,y)
is substituted for β, this yields the
y
formula
I - ( A <p(X, y) => V ψ(χ, y))
χ
<=>
y
Λ ψ( > y)
χ
χ
ν
χ
V ψ( > y))>
y
which is an example of a functional tautology. This is so because for
any propositional functions φ(χ, y), ψ(χ, y), x, y eX, the propositional
function in the form given above is true in XxX.
Examples of important functional tautologies which cannot be
obtained by application of Theorem 4.1 will now be given below.
2
The law called dictum de omni ):
ί-Λ
(1)
1
9>00>
JVe*.
) It will always be assumed that the variables occurring in those propositional
functions range over non-empty sets.
2
) This law states that if all elements of a set X satisfy a propositional function,
then any element of that set also satisfies that propositional function. For a quantifier
4. FUNCTIONAL TAUTOLOGIES
245
Let a be any element of X. If the proposition f \ φ(χ) => φ(α) is
xeX
false, then φ(α) is a false proposition and f \ φ(χ) is a true proposixeX
tion. But / \ <p(x) is a true proposition if and only if the proposition
xeX
φφ) is true for every b eX. This contradicts the assumption that the
proposition <p(a) is false. Hence every element a e X satisfies the propositional function (1), which proves that (1) is a functional tautology.
The following law will now be proved *):
hA^w^V^)-
(2)
xeX
xeX
Assume that implication (2) is false. If this be so, then \J
φ(χ)
xsX
is a false proposition and / \ φ(χ)
is a true proposition. This and
xeX
formulas
{χ eX:
(8) and
(5) in Section
1 yield {x e Χ:
φ(χ)} = Ο
and
φ(χ)} = X. Hence Χ = Ο, which contradicts the assumption
that Χ Φ Ο (see footnote *), p . 244).
2
The following law (3) h o l d s ) :
h-Ky)=>V?(*)>
(3)
xeX
y* x
Let a be any element of X. If proposition φ(α) => \J φ(χ) is false,
xeX
then <p(a) is true, while \J φ(χ)
is false, so that {x e Χ: φ(χ)} = Ο
xeX
holds. This contradicts the assumption that a satisfies ^ ( x ) . Hence,
with its range restricted by a propositional function y(x), x e X, we have the following law instead of (1):
Λ Ψ (χ) => (νϋ') => <p(y)),
ye Χ.
ψ(χ)
ι
) For quantifiers with ranges restricted by a propositional function ψ(χ), xe
we have the following law instead of (2) :
X,
Α <ρ( => ( V vw => V ΨΜ) ·
χ)
2
) For a quantifier with its range restricted by a propositional function
xe X, we have the following law instead of (3):
v(*)
ψ(χ),
ΧΠΙ.
246
FUNCTIONAL CALCULUS AND ITS APPLICATIONS
for every a e X, a satisfies (3), which means that (3) is a functional
tautology.
De Morgan's laws :
I
(4)
\J
<p(x)oA~<P(x)>
xeX
(5)
xeX
Λ v ( * ) < » \ / ~ ¥>(*)·
I
xeX
xeX
The proposition ~ \ / <p(x) is true if and only if the proposition
xeX
\J φ(χ) is false, i.e., if and only if {x eX:
<p(x)} = Ο (see Section 1,
xeX
formula (8)). This last condition is equivalent to the condition — {xeX:
ψ(χ)} = X, but, by Theorem 2.3, Chapter XII, this equation holds
if and only if {xeX:
~ φ(χ)} = X, i.e., if and only if the proposition / \ ~<p(x) is true (see Section 1, formula (5)). This proves equixeX
valence (4).
T o prove (5) note that the proposition ^ / \ φ(χ)
is true if and
xeX
only if the proposition / \ φ(χ) is false, i.e., if and only if {x e l : <p(x)}
xeX
Φ X (see Section 1, formula (7)). This condition is equivalent to the
condition —{xeX: <p(x)} Φ —X= O. By applying Theorem 2.3,
Chapter XII, we infer that the last condition is satisfied if and only
if {x e l : ~ <p(x)} Φ Ο, i.e., if and only if the proposition \/ ^ <p(x)
is true. This proves equivalence (5).
N o w (4) and (5) yield the following formulas:
(6)
~\/<p(x)
xeX
(7)
~
=
Λ~ΨΜ>
xeX
/\φ(χ)=
xeX
xeX
\/~φ(χ).
xeX
When the law of double negation (Chapter XII, Section 12, formulas
(8) and (32)) and Theorem 3.1 are applied, (6) and (7) yield
(8)
~\/~<K*) = Λ
xeX
(9)
~ / \ ~ φ { χ )
xeX
xeX
= V ?»(*)·
xeX
4. FUNCTIONAL TAUTOLOGIES
247
Equivalences (8) a n d (9) make it possible t o define the universal quantifier in terms of the existential quantifier a n d negation, a n d t o define
the existential quantifier in terms of the universal quantifier a n d negation.
Tautologies ( l ) - ( 5 ) and equivalences (6)-(9) can easily be generalized so as t o cover propositional functions of many variables. F o r
instance, the law called dictum de omni then takes on the form
(ίο)
ι- Λ
χ
φ( ι>
···>**),
···>*») => <p(y,χι,
where y eXl9
x2 GX2, . . . , xn e J n .
The remaining tautologies and equivalences are generalized similarly.
The easy proofs are left t o the reader.
Tautologies (4) and (5) and equivalences (6)-(9) can also be generalized so as t o cover those cases in which the quantifiers have their ranges
restricted by propositional functions (see Section 2). T h e easy proofs
are left t o the reader as exercises.
Examples
(I) By (6) we obtain the following equivalences:
- V (x + 1 = 0) = Λ - ( + 1 = 0 ) ,
2
χ2
xe&
xedt
~ V
Λ
(ϊ=Ψ)=
ΟΦΛςζΧ
~ ( J = F ) .
ΟφΑ<^Χ
(II) By (7), formula (29) in Chapter XII, Section 12, a n d Theorem 3.1,
we obtain the following equivalences:
~Λ
Λ (
xeM ye&
χ
<y
2χ
2
<y )
= V~Λ(
Χ
xe&
<y
2χ
2
<y )
ye&
2
<y )).
T o conclude, it is worth emphasizing that equivalences
(6)-(9),
SE ν ν ( * < > Ά ~ ( χ
2
x€& yeM
which result from D e Morgan's laws, have many applications in apagogic proofs. F o r instance, equivalence (8) is used in proofs of theorems
which assert that every element x of a set X satisfies the propositional
function φ(χ)9 χ e l . It is then assumed that there is an element χ eX
which does n o t satisfy φ(χ), that is, that the proposition
\J
xeX
~φ(χ)
248
ΧΙΠ. FUNCTIONAL CALCULUS AND ITS APPLICATIONS
is true. Starting from this assumption we try to arrive at a contradiction. By applying the rule of inference
α =>
(βΛ~β)
^
ce
we infer that the proposition ~ \J ~ φ(χ) is true. Hence, by applyxeX
ing (8), we obtain /\φ(χ).
Equivalence (6) is used in a similar way
xeX
in proofs of theorems asserting that every element χ of a set X does
not satisfy a propositional function φ(χ), x e Χ.
Equivalence (9) is often used in apagogic proofs of what are called
existential theorems, i.e., theorems which state that a mathematical object
t h a t satisfies a propositional function does exist. T o prove a theorem
in the form \J φ(χ) it is assumed that the proposition / \ ^ φ(χ)
xeX
xeX
is t r u e ; we next try t o arrive at a contradiction. Under the rule of inference
« =>
(βΛ~β)
~ OL
we infer that the proposition ~ f\ ~ φ(χ) is true. This a n d (9) yield
xeX
7
\y φ(χ). Equivalence (7) is being used analogously in apagogic proofs
xeX
of theorems which state that in a fixed X there is an element χ that
does n o t satisfy a propositional function φ(χ). Those proofs of existential theorems which are based on equivalences (9) or (7) are regarded
as non-effective proofs, since in the proof the object that satisfies (or
does not satisfy) a given propositional function is not indicated and
the method of constructing such an object is not given either. Hence
it may happen that, even though a theorem which states that there
is an object which satisfies (or does not satisfy) a given propositional
function has been proved, we are unable t o give an example of such
an object.
The proof of the existence of transcendental numbers, based on
Theorem 3.2, Chapter VIII, stating that the set 0t of all real numbers
is non-enumerable, and on Theorem 2.12, Chapter VIII, stating that
the set of all algebraic numbers is enumerable, can serve as an example
5. INTRODUCTION AND ELIMINATION LAWS FOR QUANTIFIERS
249
of a non-effective proof of an existential theorem. In accordance with
what has been said above we assume that every real number is algebraic,
that is, that it is not transcendental. By the theorems mentioned above
this assumption results in a contradiction. Hence it is not true that
every real number is algebraic (i.e., is not transcendental). F r o m (9)
we infer that there is a transcendental real number.
5. Introduction and elimination laws for quantifiers
5 . 1 . If ψ is a proposition or a propositional function not containing
any free occurrence of x, then the formulas listed below are functional
tautologies *):
(1)
h A ( ? ( * ) ν ψ) ο (Λ Ψ(Χ) ν ψ),
xeX
(2)
I-
V (φ(χ) V ψ) ο ( χ/ ψ(χ) ν ψ),
xeX
(3)
(5)
χ
Λ
ψ) >
xeX
V (φ(χ) Λ ψ) ο (xeX
V φ(χ) Λ ψ),
ι- Λ (?>(*) => ψ) ο ( V ψ w => ψ) >
h
xeX
xeX
(6)
xeX
I- A (<p(x) Α ψ) ο ( Α ψ( )
xeX
(4)
xeX
h' V (φ(χ) ^ψ)ο(Α
xeX
φ(χ)^ψ)>
(7)
(8)
Tautologies (l)-(8) will be called introduction and elimination laws
for quantifiers. They can easily be generalized so as to cover propositional functions of many variables. For simplicity proofs will be given
here for propositional functions of one variable.
T o prove (1) consider two cases. Assume first that ψ is a proposi*) Laws (1), (4), (5), (7) can be generalized so as to cover quantifiers with ranges
restricted by a propositional function. To generalize laws (2), (3), (6), (8) so as to
cover quantifiers with ranges restricted by a propositional function σ(χ), xe X, we
have to make the additional assumption that {xe Χ: σ(χ)} Φ Ο.
250
XIII. FUNCTIONAL CALCULUS AND ITS APPLICATIONS
tion. If ψ is a true proposition, then both sides of equivalence (1) are
true. The left-hand side is a true proposition since φ(α) ν ψ is true for
every aeX.
The right-hand side is a true proposition since one Summand of the disjunction is a true proposition. If ψ is a false proposition, then the left-hand side is a true proposition if and only if the
proposition φ(ά) ν ψ is true for every a e X, i.e., if and only if the p r o position φ(α) is true. The same condition is necessary and sufficient
for the right-hand side of (1) to be a true proposition under the assumption that ψ is false. We have thus proved that the left and the
right-hand sides of (1) always have the same logical value, which proves
in turn that (1) is a functional tautology.
Assume now that ψ is a propositional function w(y), y eY. An element b e Y satisfies the propositional function
only if the proposition
if and
, if and only if the
proposition φ(α) ν \p(b) is true for every a eX. This condition is
satisfied if %p(b) is a true proposition or if φ(α) is a true proposition for
every aeX.
This condition is necessary and sufficient for
to be a true proposition, and hence for b e Y to satisfy the p'ropositional function
Thus, b e Y satisfies the right-hand
side of equivalence (1) if and only if it satisfies the left-hand side.
This proves that (1) is a functional tautology.
We show in a similar way that equivalence (3) is a functional tautology. Assume that ψ is a proposition. The left-hand side of (3) is a true
proposition if and only if the proposition φ(α)Λψ is true for every
a eX, i.e., if and only if ψ is a true proposition, and φ(ά) is a true p r o position for every a eX. This is also a necessary and sufficient condition for the right-hand side of (3) to be a true proposition. If ψ is
a propositional function ip(y)9 y eY, then any element b eY satisfies
the left-hand side of (3) if and only if the proposition φ(ά)Αψφ)
is
true for every a eX, that is, if and only if \p(b) is a true proposition,
and φ(α) is a true proposition for every a eX. This is also a necessary
and sufficient condition for b to satisfy the propositional function
5. INTRODUCTION AND ELIMINATION LAWS FOR QUANTIFIERS
251
We have thus proved that equivalence ( 3 ) is a functional tautology.
The following formulas result from equivalences ( 1 ) and ( 3 ) as
proved above:
A ( P W V ? ) S ( A P W V V) ,
(ΐ')
xeX
xeX
A(?WA?)S(APWA?).
(3')
xeX
xeX
On applying formula (9), Section 4 ; formula (18), Chapter X I I , Section 1 2 ; Theorem 3 . 1 ; Theorem 5.2, Chapter X I I ; ( 3 0 ; formula (17),
Chapter X I I , Section 1 2 ; formula (31), Chapter X I I , Section 12, we
obtain
0(*)
ν ψ) = — A ~ (<P(x)
xéX
v
ψ)
xeX
= ~ xeX
/\ (Α φ(χ) A ~ ψ) = ~ (A
^ φ(χ)
xeX
= l\r^j φ(χ) ν ^ ~ψ = \f φ(χ) V ψ.
xeX
Α^ψ)
xeX
Hence
(20
VfrW ?) (V>Wvv),
v
5
xeX
xeX
which proves that (2) is a functional tautology.
On applying formula (9), Section 4 ; formula (17), Chapter X I I ,
Section 1 2 ; Theorem 3 . 1 ; Theorem 5.2, Chapter X I I ; ( Γ ) ; formula
(18), Chapter X I I , Section 1 2 ; formula (31), Chapter X I I , Section 12,
we obtain
V (φ(χ) Α ψ) = ~ A
xeX
χ Λ
~ (φ( )
Ψ)
xeX
= ~ xeX
A (~
φ(χ) V ^ ψ) = ~ (A ^ φ(χ) V
"
'
xéX
=
/ \
ψ(χ)
xeX
A
=
\ /
ψ(χ)
xeX
Hence
(40
V(?(*)AV) s
(VPWAV),
which proves that (4) is a functional tautology.
.
Λ ψ .
~ψ)
252
FUNCTIONAL CALCULUS AND ITS APPLICATIONS
Xlll.
On applying formula (14), Chapter XII, Section 12; Theorem 3.1;
( Γ ) ; formula (6), Section 4 ; Theorem 5.2, Chapter XII, we obtain
Λ (φ(χ) => ψ) = Λ (~ φ(χ) ^ψ) = (Λ ~ φ(χ)
xeX
xeX
ν
ψ)
xeX
= ~ V <ρ( ) w = V ψ( ) => ψχ
v
χ
xeX
Hence
xeX
Λ (ψ( ) => νθ (V ψ( ) => ψ),
ξ
χ
(50
χ
xeX
xeX
which proves that (5) is a functional tautology.
On applying formula (14), Chapter XII, Section 12; Theorem 3.1;
(2'); formula (7), Section 4 ; Theorem 5.2, Chapter XII, we obtain
xeX
χ ν
χ v
χ
V (<ρ( ) => ψ) = V (~
Ψ) = V ~ ψ( )
ψ( )
xeX
ψ
xeX
= ~ Λ ψ( )
χ ν
ψ =
A
ν·
Hence
V
(6')
V)
s
(Λ
V).
which proves that (6) is a functional tautology.
On applying formula (14), Chapter XII, Section 12; Theorem 3.1;
formula (33), Chapter XII, Section 12; Theorem 5.2, Chapter X I I ;
(Γ), we obtain
Λ (ψ => φ(χ)) = Λ (~ ψ (*)) = Λ
ν
xeX
xeX
ΞΞ
ν
~ Ψ)
xeX
A (χ) ν ~ ψ = ~ ψν / \ φ (χ) = ψ => f \ φ (χ).
99
.xeX
xeX
xeX
Hence
(7')
Λ ( ν =>?(*)) = (ν => A
ΛΈΛ"
xeX
which proves that (7) is a functional tautology.
On applying formula (14), Chapter XII, Section 12; Theorem 3.1;
formula (33), Chapter XII, Section 12; Theorem 5.2, Chapter X I I ;
(2'), we obtain
5. INTRODUCTION AND ELIMINATION LAWS FOR QUANTIFIERS
253
Hence
(80
which proves that (8) is a functional tautology.
Tautologies (2), (4)-(8) could also be proved directly, as has been
done in the case of (1) and (3). Equivalences ( Γ ) - ( 4 ' ) make it possible
to introduce a quantifier that precedes a disjunction or a conjunction
into one component on the condition that the other component does
not contain any free occurrence of a variable which is bound by that
quantifier. These equivalences also make it possible to eliminate a quantifier from a component of a disjunction or a conjunction and to place
it before t h a t disjunction or conjunction as a whole, on the condition
that the other component does not contain any free occurrence of a
variable which that quantifier would then bind.
Equivalences (5')-(8') make it possible to introduce a quantifier
preceding an implication into the consequent of that implication,
on the condition that the antecedent does not contain any free occurrence of a variable bound by that quantifier; they also make it possible
to introduce a universal quantifier preceding an implication into the
antecedent of that implication while changing it into an existential
quantifier in the process and to introduce an existential quantifier
preceding an implication into the antecedent of that implication while
changing it into a universal quantifier in the process, on the condition
that the consequent of that implication does not contain any free
occurrence of a variable bound by that quantifier. Equivalences (5')(8') further enable us to eliminate quantifiers from the antecedent of
an implication to the position preceding the whole implication, while
changing a universal quantifier into an existential one, and vice versa,
in the process, and also to eliminate quantifiers from the consequent
of an implication to the position preceding the whole implication;
the conditions that the other component of the implication in question
does not contain any free occurrence of a variable which the quantifier
in question would then bind, must be satisfied, respectively.
254
XllL FUNCTIONAL CALCULUS AND ITS APPLICATIONS
Tautologies (5)-(8) and the resulting equivalences (5')-(8') may
seem not quite intuitive. This is why examples that illustrate some applications of these equivalences are given below; it is hoped that they
will help the reader to grasp the meaning of these formulas.
Examples
(I) The following propositional function is true in the set Jf
all natural numbers :
(9)
of
Λ ^ m => η = 1.
n
By (5'), this propositional function is equivalent to the following one:
V
(n < m => η = 1).
Hence the above propositional function is also true. In fact, for every
natural number η there is an m9 namely m = 1, such that the implication: if η < m, then η = 1, is true.
(II) Consider now the implication converse to (9). It is also a propositional function true in Jf:
(10)
η = 1 => Λ η < m.
meJf
By (7'), (10) is equivalent to the following propositional
which is also true in Jf :
function,
A (n = 1 => η < m).
(III) Consider the following propositional function, which is true
in^yT:
(11)
V m+k =
n=>m<n.
Under (6') we infer that the following propositional function is equivalent to (11) and hence also true in Jf\
A (m + k = η => m < η).
(IV) Consider now the implication converse to (11), which also is
true in Jf:
(12)
m < η => V m + k = η.
6. DISTRIBUTIVITY LAWS FOR QUANTIFIERS
255
On applying equivalence (8') we infer that (12) is equivalent to the following propositional function, also true ί η . / Γ :
V (m < η => m + k = ή).
In fact, the above propositional function is true, because for any natural numbers m, η there is a k eJf such that the implication: if m < n,
then m + k = n, is true.
6. Distributivity laws for quantifiers
In this section we discuss laws in the functional calculus which
apply to the relationships between the following formulas:
Α (φ (χ) ° ψ(χ))
Α φ (χ) ° A w(x)
and
xeX
xeX
xeX
and
V (φ (x) ° ψ(χ))
V ψ (x) ° V
and
xeX
xeX
ψ(χ),
xeX
where the symbol ο will be replaced by the symbols of disjunction,
conjunction, and implication.
The law of distributivity of the universal quantifier over conjunction *)
(l)
I - Λ (φ(χ) Α ψ(χ)) ο ( Α ψ(χ) Λ Α
xeX
xeX
ψ(χ)).
xeX
The proposition on the left-hand side of (1) is true if and only if
X = {x eX:
φ(χ) Αψ(χ)}
(see formula (5), Section 1). U n d e r Theorem 2.1, Chapter XII, this
condition is equivalent t o the following o n e :
X = {xeX:
φ(χ)} n{x e Χ:
ψ(χ)}.
The above equation holds if and only if
{x G X: <p(x)} = X
and
{x G Χ: ψ(χ)} = Χ,
*) Law (1) has a simple interpretation in the algebra of sets. It states that the
intersection of any two subsets of a fixed space X equals X if and only if each of these
subsets equals the whole space X.
XIII. FUNCTIONAL CALCULUS AND ITS APPLICATIONS
256
that is, if and only if the proposition A <p(x) is true and the proposixeX
tion Α ψ(χ) is
t r u e
e
> i- -> if
a n
d
o n
ty if
t ne
proposition on the right-hand
χεΧ
side of (1) is true. This proves that (1) is a functional tautology.
Tautology (1) can be generalized so as to cover the case of propositional functions of many variables and the case of quantifiers with
ranges restricted by a propositional function. The simple proofs are
left to the reader.
The universal quantifier is not distributive over disjunction, which
means that if in (1) the symbol Λ is replaced by the symbol ν , then
the resulting formula is not a functional tautology. Consider the following two propositional functions : χ < 0, χ e 0t, and χ > 0, χ e 0i
(where ^ is the set of all real numbers). The proposition
ν χ > 0) is true since for every real number a the proposition a
ν a > 0 is true. But the proposition
the proposition
0 is false, since
3 is false and so is the proposition
Hence the implication
is false. It follows therefrom that the equivalence obtained from the
above implication by the replacement of the implication symbol by the
equivalence symbol is also false. This proves that the universal quantifier is not distributive over disjunction. On the other hand, the following implication is a functional tautology *):
(2)
Assume that the consequent is false. This means that the propositional function φ(χ) ν ψ(χ) is not satisfied for every a e l , so that there
is an a e X such that φ(α) ν ψ(α) is a false proposition. It follows therefrom that the proposition φ(α) is false and the proposition ψ(α) is falsel
) Law (2) has the following interpretation in the algebra of sets: if at least
one of two given subsets of a fixed space X equals the whole space, then their union
equals the whole space X.
6. DISTRIBUTIVITY LAWS FOR QUANTIFIERS
Consequently,
propositions
and
257
are
both
false,
and hence the antecedent of (2) is also false. This proves that (2) is
true. Implication (2) remains a functional tautology if φ and ψ stand
for any propositional functions of many variables. The simple proof
of this statement is left to the reader. Tautology (2) can also be generalized
so as to cover quantifiers with ranges restricted by a propositional
function.
The law of distributivity of the existential quantifier over disjuncx
tion )
(3)
Formula (6) in Section 1 and Theorem 2.2 in Chapter XII imply
that the proposition
ψ(χ)) is true if and only if
The above condition is equivalent to the following o n e :
i.e., the condition that the proposition on the right-hand side of equivalence (3) is true. This proves that (3) is a functional tautology.
Equivalence (3) remains a functional tautology if φ and ψ stand
for any propositional functions of many variables and if it contains quantifiers with ranges restricted by a propositional function.
The easy proofs are left to the reader.
The existential quantifier is not distributive over conjunction, which
means that if the symbol ν in (3) is replaced by the symbol Λ , then
the resulting formula is not a functional tautology. Consider the following two propositional functions:
(where Jf is the set of all natural numbers). The proposition
*) Law (3) has the following interpretation in the algebra of sets: the union of
any two subsets of a fixed space X is non-empty if and only if at least one of them
is non-empty.
XIII. FUNCTIONAL CALCULUS AND ITS APPLICATIONS
258
is true, since there is an even natural number and there is an odd
natural number. But the proposition
V
(2|* Λ ~ ( 2 | * ) )
is false, since there is n o natural number which is both even and odd.
These two propositions are thus not equivalent, and the implication
(V
2\x A V
xeJf
xeJT
~
(2|x)) => V
(2|ΧΛ -
(2|x))
xtJf
is false.
On the other hand, it will be proved that the following implica1
tion is a functional t a u t o l o g y ) :
(4)
Assume that the consequent of the above implication is false. Then
at least one of the propositions
is false. Hence it
follows that n o element of the set X satisfies φ(χ) or that n o element
of X satisfies ψ(χ). Thus no element of X satisfies φ(χ)Λψ(χ),
which
means that the antecedent of (4) is false. We infer that implication (4)
is true, and hence is a functional tautology.
Implication (4) remains true if φ and ψ stand for any propositional
functions of many variables and if it contains quantifiers with ranges
restricted by a propositional function. The easy proofs are left to the
reader.
The following formulas result from ( 1 ) and (3):
(1')
(3')
The law of splitting the universal quantifier over impliction
2
):
*) This law has the following interpretation in the algebra of sets: if the intersection of two subsets of a fixed space X is non-empty, then each of them is nonempty.
2
) This law has a simple interpretation in the algebra of sets : it states that if
a subset of a space X is contained in another subset of X, then if the former subset
equals X then the latter subset also equals X.
259
6. DISTRIBUTIVITY LAWS FOR QUANTIFIERS
(5)
I-
Λ OW =>
=> (Λ φ(χ) => A yW).
Assume that the consequent of implication (5) is false. Then the proposition
is true and the proposition / \ ψ(χ) is false. Thus
every element of X satisfies φ (χ) and there is an element a eX such that
ψ(ά) is a false proposition. Hence it follows that the proposition φ(α)
=> ψ(α) is false since ψ(α) is true and ψ(α) is false. Thus the element a
does not satisfy the propositional function φ(χ) => ψ(χ), which proves
that the antecedent of implication (5) is a false proposition, and implication (5) is, accordingly, true. It remains true if ψ and ψ stand
for any propositional functions of many variables and if it contains
quantifiers with ranges restricted by a propositional function. The easy
proofs are left to the reader.
The universal quantifier is not distributive over implication, which
means that the central implication symbol in (5) may not be replaced
by the equivalence symbol. This is the case because the implication
converse to (5) is not a functional tautology. Consider the following
two propositional functions :
χ < 0,
jcel,
and
> 0,
x+l
χ e 01
(where 0t stands for the set of all real numbers). The propositions
Λ x < 0
and
f\x+l
> 0
are false, and hence the implication
is a true proposition. N o w the proposition
/ \ (χ < 0=>x+\
> 0)
is false, since, for instance, — 2 < 0 , but ^ ( — 2 + 1 > 0 ) . Hence the
implication converse to (5) is false for the propositional functions
φ(χ) and ψ(χ) thus selected.
XIII. FUNCTIONAL CALCULUS AND ITS APPLICATIONS
260
The following functional tautology will be called the law of splitting the universal quantifier into existential quantifiers *):
(6)
Λ
I-
=>
=> ( V φ(χ) => V ΨΜ) ·
xeX
xeX
xeX
Assume that the consequent of implication (6) is false. Then \ /
φ(χ)
xeX
is true and \/ ψ(χ) is false. Thus there is an a e l such that φ(α) is
xeX
a true proposition and no element of X satisfies the propositional function ψ(χ). It follows that the proposition φ(α) => ψ(α) is false and thus
the antecedent of (6) is false, too. Hence implication (6) is true. Implication (6) remains true if φ and ψ stand for any propositional functions
of many variables and if it contains quantifiers with ranges restricted
by a propositional function. The proofs are left to the reader.
The existential quantifier is not distributive over implication, nor
have we any law analogous to (5).
7. Laws on the relettering and on the alternations of quantifiers
The following functional tautologies are called laws on the reletter2
ing of quantifiers ) :
(1)
I-
/ \ φ ( χ 9Χ ΐ 9
...,*„) ο
xeX
Λ <p(y,xi,
···,*„),
yeX
where xx e Xx, ..., xn e Xn,
V
(2)
-
- M ^ ^ V ^ ^ i ' ···>*»)»
xeX
yeX
where xt eXl9
1
...,xneXn.
) This law can, in the algebra of sets, be interpreted as follows: if a subset
of a space X is contained in another subset of X, then the condition that the former
subset is non-empty implies that the latter subset is also non-empty.
2
) These laws can be generalized so as to cover quantifiers with ranges restricted
by a propositional function. Law (1) then assumes the following form:
/\<p(x9Xi,
v(*)
...,Xn)o/\<P(y>Xi>
where x, y e X, Xi e Xl9 ..., xn e Xn.
Law (2) is changed similarly.
vOO
• · · » * « ) »
7. LAWS ON THE RELETTERING A N D ON THE ALTERNATIONS OF QUANTIFIERS
261
A n element ( α , ,
an) 6 ^ χ ... xXn satisfies the left-hand side
of equivalence (1) if a n d only if the proposition φ(α9 al9
an) is true
for every aeX, that is, if a n d only if the element (al9 . . . , # „ ) satisfies
the right-hand side of (1).
Likewise, an element (al9
an) eXx χ ... xX„ satisfies the lefthand side of equivalence (2) if a n d only if there is an a e X such that
the proposition φ(α9 ai9 ..., an) is true, which is equivalent t o the condition that this element satisfies the right-hand side of (2). Hence (1)
and (2) are functional tautologies.
The following formulas result from (1) a n d (2):
(V)
Λ φ(χ,χι>
···>*„) = Α ν Ο ^ ι » ···>
xeX
(Τ)
* Π ) >
yeX
V φ(χ>χι> ···>**) = V P Ö ' ^ I »
xeX
···>*,.)>
yeX
where xx eXl9
xn
eXn.
Finally, t o conclude t h e listing of the most important laws holding in the functional calculus, we give the following laws governing
the alternations of quantifiers:
(3)
f-AAf(*j)<»AAf(*j).
xeX yeY
(4)
hVVfMoVVfi'i.j').
xeX yeY
(5)
yeY xeX
yeY xeX
h V A φ(χ, y) => Λ V Φ, y)xeX yeY
yeY xeX
The left-hand side of equivalence (3) is a true proposition if a n d
only if the proposition q>(a9b) is true for every aeX a n d for every
b eY, that is, if a n d only if the proposition <p(a9 b) is true for every
b e Y and for every a e l , which is equivalent t o the condition that the
proposition on the right-hand side of (3) is true. This proves that (3)
is a functional tautology.
The proposition on the left-hand side of equivalence (4) is true
if a n d only if there are a n a e X a n d a b e Y such that the proposition
φ(α9 b) is true, that is, if and only if there are a b e Y and an a eX such
262
XIII. FUNCTIONAL CALCULUS AND ITS APPLICATIONS
that <p{a, b) is a true proposition, which is equivalent to the condition
that the proposition on the right-hand side of (4) is true. This proves
that (4) is a functional tautology.
T o prove that implication (5) is a tautology assume that its antecedent is a true proposition. Then there is an element a0 eX such that
the proposition /\φ{α0,γ)
is true, that is, the proposition
yeY
(p(a0,b)
is true for every beY. Thus for every beY there is an a = a0 eX
such that the proposition φ(α, b) is true, so that the consequent of
implication (5) is true. This proves that (5) is a functional tautology.
Equivalences (3) and (4) and implication (5) remain functional
tautologies if φ is a propositional function of m variables, where m > 2.
These tautologies can also be generalized so as to cover quantifiers
with ranges restricted by propositional functions. The easy proofs
a/e left to the reader.
Equivalences (3) and (4) yield the following formulas:
0')
Λ Λ ? Μ
xeX yeY
(4')
3
Λ Α ? Μ .
yeY xeX
V V ? ( * j ) = VVfl>(*.>oxeX yeY
yeY xeX
In (5), the implication symbol may not be replaced by that of equivalence, since the implication converse to (5) is not a functional tautology. This can be demonstrated by the following example. Let (p(x,y)
be the propositional function
x < y>
x,y e &
(where 01 is the set of all real numbers). The proposition f\\j x < y
y
χ
is true, since for every real number y there is a real number χ which is
less than y. On the contrary, the proposition \J , \ x < y is false,
x
y
as there is n o real number χ that is less than every real number y. The
implication
A y x<y=>\/
y
is thus false.
χ
f\.x<y
χ
y
8. RULES OF INFERENCE
263
8. Rules of inference
In the functional calculus, rules of inference are operations which
associate with certain finite sequences of schemata of propositional
functions (propositions), constructed from initial propositional functions by means of propositional connectives and quantifiers, a schema
of a propositional function (proposition) so that these operations,
when applied to true propositional functions (propositions), yield
true propositional functions (propositions). In the functional calculus,
rules of inference will be symbolized by means of a notation similar
to that used for rules of inference in the propositional calculus.
It follows from Theorem 4.1, Chapter XII, that in the propositional
calculus rules of inference may be treated as rules of inference in the
functional calculus. In particular, the rules that hold include: modus
ponens (rule (2), Chapter XII, Section 4); the rules of detachment
for equivalence (rules (1), Chapter XII, Section 6); the rules of
hypothetical syllogism (rules (1), (2), (3), Chapter XII, Section 8);
rules (l)-(8), Chapter XII, Section 9; rules (l)-(4), Chapter XII, Section 10; rules (1), (6), (8), (9), (12), Chapter XII, Section 11.
The rule of generalization
<p(x),
xeX
Λ Ψ(Χ)
xeX
is a rule of inference, for if φ(χ),χεΧ,
is a propositional function
true in X, then for every a e X the proposition φ(α) is true, which
proves that the proposition / \ φ{χ) is true. Rule (1) is also applicable
xeX
to quantifiers with ranges restricted by any propositional function
ψ(χ), x e X.
The law of dictum de omni (formula (1), Section 4) and the rule of
1
modus ponens yield the following rule of inference ):
l
ψ(χ),
) In the case of a quantifier with its range restricted by a propositional function
xe X, we have the following rules instead of (2):
Λ φ(χ)
Λ φ(χ)
y**)
ψΟΟ
ψ(γ) => 9>0), ye Χ
9
φ(γ), y e {χ e Χ: ψ(χ)}
264
ΧΠΙ.
FUNCTIONAL CALCULUS AND ITS APPLICATIONS
(2)
XEX
<p(y),yeX
χ
This is the case because if the proposition / \ ψ( )
xeX
*
s
t r u e
> then, on
applying the rule of modus ponens to f \ φ(χ) and to / \ φ(χ) => φ{γ),
xeX
xeX
y e X, we obtain (p(y), y e X.
In a similar way, tautology (2), Section 4, and tautology (3), Sec1
tion 4, yield the following two rules of inference ):
(3)
—
,
xeX
Tautology (2), Section 6, tautology (4), Section 6, and the rule
of modus ponens yield the following two rules of inference:
/ \ ψ(χ) ν / \ y(x)
(5)
/ \ (φ(χ) ν ψ(χ))
^ contracting the universal quantifier
over disjunction,
xeX
χ
V (ψ( )
(6)
Λ
χ
ψ( ))
\J φ(χ) A V ψ(χ)
xeX
ir eu j>
0
Spiitting
the existential
quantifier
over conjunction.
xgX
These rules are also applicable to quantifiers with ranges restricted
by a propositional function σ(χ)9 x e Χ.
The law of splitting the universal quantifier over implication, formula
(5), Section 6, the law of splitting the universal quantifier into existential
J
) Rule (3) may be applied to a quantifier with its range restricted by a propositional function ψ(χ), x e X, if and only if {x e Χ: ψ(χ)} Φ Ο. Instead of (4) we have
the following rule :
V>(y)A(p(y),ye X
V <p(x)
ψ(χ)
8. RULES OF INFERENCE
265
quantifiers, formula (6), Section 6, a n d the rule of modus
yield the following two rules:
ponens
Λ (φ(χ) => v(*))
Α φ( ) => A
Α OK*) => ¥>(*))
xeX
(7)
χ
(8)
~\~~T~
—,
xeA"
xeX
Rules (7) a n d (8) are also applicable t o quantifiers with ranges restricted
by a propositional function.
The law (formula (5), Section 7) of alternation of quantifiers
and the rule of modus ponens yield the following rule of alternation of
quantifiers, which also is applicable t o quantifiers with ranges restricted
by propositional functions:
\f
/Q\
f\<p(x,y)
xeX yeY
yeY xeX
The rule of introducing the universal
quantifier:
ψ=>ψ(Χ)9ΧΕΧ
(10)
φ
=> Α ψ( )
χ
xeX
holds under the assumption that φ does not contain any free occurrence
of x. Assume that φ => ψ(χ), x e X, is a true propositional function.
By applying the rule of generalization (1) we obtain / \
xeX
(φ=>ψ(χ)).
But, by formula (7'), Section 5, the last formula is equivalent t o
ψ => /\ ψ(χ)9 which proves that (10) is a rule of inference. It is also
xeX
applicable t o quantifiers with ranges restricted by a propositional
function σ(χ)9 x e Χ.
The rule of introducing the existential quantifier:
(υ)
φ(χ)=>ψ,χβΧ
V
φ(χ) => w
xeX
>CIIT. FUNCTIONAL CALCULUS AND ITS APPLICATIONS
266
holds under the assumption that no free occurrence of χ appears in ψ.
Assume
that
φ(χ)=>ψ,
xeX,
is a
true propositional
function.
By applying the rule of generalization (1) we obtain / \ (φ(χ) => ψ).
xeX
By (5'), Section 5, the last formula is equivalent to
v
xeX
φ(χ) => ψ, which
proves that (11) is a rule of inference. It is also applicable to quantifiers
with ranges restricted by a propositional function, σ(χ), χ e Χ.
l
The rule of eliminating the universal quantifier ):
ψ =>
A ψ(χ)
X
(12)
x e
φ => ψ(χ)9 χ e Χ
If φ => / \ ψ(χ) is true, then, since / \ ψ(χ) => ψ(χ), χ e X (see formula
xeX
xeX
(1), Section 4), is true, on applying the rule of hypothetical syllogism
(see formula (1), Chapter XII, Section 8) we obtain φ => ψ(χ), χ e Χ.
l
The rule of eliminating the existential quantifier ):
(13)
By (3), Section 4, the propositional function <p(x) =>
is true. If the proposition
χ e X,
is true, then on applying the
rule of hypothetical syllogism (see formula (1), Chapter XII, Section 8)
we obtain φ(χ) => ψ, χ e X. This proves that (13) is a rule of inference.
All the rules of inference given above are applicable to propositional
functions of many variables.
The search for laws of the functional calculus is much more difficult
than in the case of the propositional calculus. This is due to the fact that
') For a quantifier with its range restricted by a propositional function σ ( χ ) ,
xe X, instead of (12) we have the following rule:
Instead of (13) we have the following rule:
9. QUANTIFIERS VERSUS GENERALIZED UNIONS AND INTERSECTIONS OF SETS
267
a proposition can take on two logical values only, i.e., truth and falsehood, whereas propositional functions can be satisfied by certain elements in the ranges of the variables occurring in those functions, but may
not be satisfied by other elements. The variables which occur in propositional functions usually range over infinite sets. Hence it is not to be
expected that a method can be found which, like the truth-table
verification method for the propositional calculus, would enable one
to single out functional tautologies. In fact, it has turned out that the
functional calculus, unlike the propositional calculus, is undecidable,
which means that there is no method which would enable one to
decide, in a finite number of operations, whether a given schema of a
propositional function is, or is not, a functional tautology. We have
been successful in doing this in certain simple cases by analysing the
meanings which we ascribe to propositional connectives and quantifiers.
This method fails in more complicated cases.
New tautologies can be obtained by rules of inference from known
functional tautologies. This fact suggests the idea of applying the
axiomatic method to the functional calculus *). Certain tautologies are
then adopted as the initial ones (axioms), and certain rules of inference
also are adopted as initial ones. Other tautologies are obtained from
axioms by rules of inference. The functional tautologies and rules
of inference given in the present book cover the axioms and the rules
of inference adopted in the axiomatic systems of the functional calculus and form a logical apparatus which suffices in applications of the
functional calculus in proofs of mathematical theorems.
9. Quantifiers versus generalized unions and intersections of sets
Equivalences (1) and (6), Chapter IV, Section 1, express the relationships between the concept of generalized union of sets and the
existential quantifier, and the concept of generalized intersection of
sets and the universal quantifier. When proving the properties of generalized unions and intersections of sets in Chapter IV, Section 2, we did
l
) An axiomatic approach to the functional calculus is described, e.g., by
S. C. Kleene, Mathematical logic, New York 1967.
268
XIII. FUNCTIONAL CALCULUS AND ITS APPLICATIONS
not refer to functional tautologies, which figured there in a disguised
manner. Those proofs become much clearer and more precise when
carried out more formally, with reference to now known tautologies
in the propositional and in the functional calculus.
Examples
(I) We shall now prove equation (4), Theorem 2.3, Chapter IV.
Formula (1), Chapter I, Section 2 ; formula (1), Chapter IV, Section 1;
formula (32), Chapter XII, Section 12; formula (2'), Section 5, yield
The equation
is an analogue, in the algebra of sets, of the following functional tautology (see formula (2), Section 5):
(II) D e Morgan's law (16), Chapter IV, Theorem 2.7, will now be
proved. By formula (1), Chapter I, Section 5; formula (1), Chapter IV,
Section 1 ; formula (6), Chapter IV, Section 1 ; formula (6), Section 4,
we obtain
De Morgan's law referred to above is an analogue, in the algebra of sets,
of De Morgan's law in the functional calculus (formula (4), Section 4).
Other properties of generalized unions and intersections of sets are
proved similarly. It is worth noting that the equations in Theorem 2.3,
Chapter IV, are analogues of the functional tautologies listed as formulas (2), (4), (1), (3) in Theorem 5.1; that the equations in Theorem
2.4, Chapter IV, are analogues of the functional tautologies listed
as formulas (3) and (1) in Section 6; that the set inclusions in Theorem
9. QUANTIFIERS VERSUS GENERALIZED UNIONS A N D INTERSECTIONS OF SETS
269
2.5, Chapter IV, are analogues of the functional tautologies listed as
formulas (4) and (2) in Section 6; and that D e Morgan's laws (16)
and (17) in Theorem 2.7, Chapter IV, are analogues of De Morgan's
laws in the functional calculus, listed as formulas (4) and (5) in Section 4.
Let now q>(x,y), x,yeX,
be any propositional function of two
variables. If that propositional function is prefixed with the quantifier
or the quantifier
functions
then we obtain the following propositional
of one variable
Consider
now the following two sets:
(1)
(2)
On the other hand, for every fixed v 0 e 7 we can form a set
In this way we obtain the indexed family of sets
The following theorem establishes the relationship between the
concepts of generalized union and intersection of sets and those of
quantifiers:
9.1. For any propositional function φ(χ, y), χ e X, y e Y,
(3)
(4)
T o prove (3) note that the following equivalences hold:
These equivalences yield (3). Equation (4) is proved similarly.
270
XIII. FUNCTIONAL CALCULUS AND ITS APPLICATIONS
Examples
(III) Let $ stand for the set of real numbers, and Q for the set
of rational numbers. Consider the following two propositional functions:
2
x
< w,
χ Ε 01,
we O,
3
and
w < χ,
χ e 01,
w e Q.
Let us form two indexed families of sets:
(AW)WS2
= ({Χ E 0l\ X < W}) $ ,
2
(BW)WG2
WE
= ({Χ Ε 0l\ W < X }) 2
3
WE
Our task will be to find the generalized union U (AwnBw).
On ap-
plying formula (4) and Theorem 2.1, Chapter XII, we obtain
(5)
2
= U ({x e 0l\ x < w}n{xe
U (AwnBw)
3
0l\ w < x })
= U {x ^ 0t\ X < WAW < X }
2
3
= { x e l : \J (x < wAW <
2
3
x )}.
As we know,
\J (X < W AW < Χ ) = X < X .
2
3
2
3
This and (5) yield
U (AwnBw)
2
3
= {x e 0t: x < x } = {x e 0t\ I < x}.
(IV) Let 0t stand for the set of all real numbers, and Jf for the
set of all natural numbers. Let ( / n ) n G^ r be a sequence of functions
defined on 0t and with values in 0t, uniformly convergent t o a function / : ^2 -> 0t. Consider the following propositional function
φ(χ, m, k, ή), χ Ε 0t, m,n,k
eJf:
(6)
n> k^\fn(x)-f(x)\
<l/m.
Let us form a triply indexed family of sets (Amkn)mtkinejr
r
r
r
xJ
for every (m,k,ri) eJ xJ
Amkn
= {xe 0l\ n> £ =>\f n(x)-f(x)\
We show that
00
00
« = n u
00
C\AmUn.
<
by defining
l/m}.
.
10. EXAMPLES OF APPLICATIONS OF THE FUNCTIONAL CALCULUS
271
00 00 00
Π U Π Amkn c 0t, it suffices to show that the set 0t is conm=l /c=l n--=l
tained in the set on the right-hand side of the equation above. By
Theorem 9.1 we have
00 00 00
CO OO 00
Since
n u n
= 1k
= n u n { x e i : «
Amkn
m= 1 k= 1 «
1 η= 1
> *=>I/.(*)-/WI <
1
= { . v e l : Λ V Λ (« > Λ => \f„(x)-Äx)\
m
k
<
lIm)}.
η
Since by assumption the sequence (f„)nejr
is uniformly convergent to
the function / , the following propositional function is satisfied for
every χ e 0t\
Λ V A (n>k=>
m
k
\fn{x)-f(x)\
< 1/m).
η
00 00 00
This proves that 0i a (~) U P | ^ m J t n, which was to be demonstrated.
10. Examples of applications of the functional calculus in mathematical
proofs
Proving mathematical theorems consists in successive simple arguments in each of which a proposition or a propositional function is accepted as a direct logical consequence of other propositions or propositional functions, whose truth was proved or adopted by an earlier convention. Each step in such an argument is based on a functional tautology
or a rule of inference. A number of simple examples of applications
of the functional calculus in mathematical proofs will be given below.
Examples
be any sequence of real numbers. This sequence
(I) Let (a„)„ejr
is convergent if there is a real number a which is the limit of that sequence, which in symbols will be written t h u s :
(an) is convergent ο V Λ V
α ε>0
k
/ \ (η > k => \an-a\
< ε),
η
where a and ε range over the set of real numbers, and k and n, over
272
XIII. FUNCTIONAL CALCULUS AND ITS APPLICATIONS
the set of natural numbers. This shows that the condition of convergence
is expressed by a formula whose logical structure is fairly complex.
By Theorems 5.2 and 3.1, Chapter X I I ; formulas (6) and (7), Section 4;
and formula (29), Chapter XII, Section 12, we obtain
(an) is not convergent <
This holds because
Thus, if we are to form a proposition or propositional function equivalent to the negation of a proposition or propositional function that
begins with quantifiers, every universal quantifier is to be replaced by an
existential one, every existential quantifier is to be replaced by a universal one, and after the quantifier we have to write the negation of the
formula which originally followed the quantifiers. The laws of the propositional calculus show that if that formula is an implication α => β,
then its negation is equivalent to the conjunction OLA ~ β. Let us apply
this method to example (II) in Section 3. We then obtain
(II) The condition of discontinuity of a function / : 0t
ae 0t\
M at a point
f is discontinuous at a <
(III) Consider example (IV), Section 3. It provides a formulation, in
logical symbolism, of the principle of induction. On applying equivalences
(3), Chapter XII, Section 7; (17) and (29), Chapter XII, Section 12;
and (4), Section 7, to the formula within the scope of the quantifier
273
10. EXAMPLES OF APPLICATIONS OF THE FUNCTIONAL CALCULUS
/\
we obtain the following proposition, which is equivalent t o the
principle of induction :
( V » M = > ( l M v
Λ
V
(keAAk+l
k + l eAj)=>
/ \ η ΕA
φΑ))).
This is the case because
(IEAA
f\
(k EA=>
kzJf
neJf
= ~ /\ΠΕΑ=>
~(l
EAA
EÄ)).
f\(kEA^k+\
Also
~
~ ( 1 EAA
/ \ η Ε A = \/
Λ (kEA^>k+\EÂ))
ηφΑ,
= (\tA\<
V/ (kEAAk+l
φΑ)).
(IV) A function / : 0t
0t is uniformly continuous on the set 0t of
real numbers if and only if the following condition is satisfied :
(1)
Λ
V Λ Λ ( Ι * ι - Χ ι \ < à => [ / " ( * , ) <
ε).
e>0 <5>0 Λ"ι x2
By formula (5), Section 7, condition (1) implies the condition
(2)
Λ Λ V Λ (l^i -xi\ < » => Ι/(*ι)-/(* 2 )Ι < e).
ε>0 χι <5>0 χ2
By formula (3'), Section 7, condition (2) is equivalent to the condition
(3)
Λ Λ
V Λ (Ι*ι - * 2 \ < à =>\f(Xl)-Äx2)\
<
ή.
χι ε>0<5>0 χ2
Condition (3) is the condition of the continuity of a function / on the
set of real numbers. We have thus proved that uniform continuity of
a function implies the continuity of that function. It can be proved
similarly that uniform convergence of a sequence of functions implies
convergence in the ordinary sense of the term.
(V) Let Χ Φ Ο be any subset of the set 0t of real numbers. A number
a Ε 0t is called the least upper bound of X, written : a = sup X, if and
only if the following condition is satisfied :
(4)
Λ (* - α) Λ (Λ (* < y) => (« < y)) •
Λ
xeX
ye& xeX
274
XIII. FUNCTIONAL CALCULUS AND ITS APPLICATIONS
By formula (3), Chapter XII, Section 7, and formula (7), Section 4, we
obtain the following equivalences:
/ \ (x < y) => ifl < y) = ~ iß < y)=> ~ f\(x
xeX
)y
xeX
< a=> \/
= y
y
X-
<
xeX
This, Theorem 3.1, and Theorem 5.2, Chapter XII show that condition
(4) is equivalent to the following one:
Λ (χ - ί à) Λ (ϋ> < α) => V (y < χ)).
(40
Λ
xeX
ye9i
xeX
We have thus obtained another equivalent definition of the least upper
bound of any subset Χ Φ Ο of the set of real numbers.
(VI) Let (a„)neJr be any sequence of real numbers. We prove the
following theorem:
(5)
lime» =
α<>Λ V Λ \a„-a\
n->ao
m
no
<
l/m,
n>n0
G / , a G 0t, (an)e
where m,n0,n
J
0t '\
It follows from the definition of the limit of a sequence that the following propositional function is true:
(6)
Y\man = a=>/\
\J
π-+οο
wo n>riQ
ε>0
f\
\an-a\
< ε,
e e l
Hence, by applying equivalence (7'), Section 5, we obtain the following
propositional function which is equivalent to (6) :
A ( l i m e , = a=>\J Λ \a -a\
n
e>0
π —• oo
<
^
w
m
1
riQ n>fiQ
On applying rule (2), Section 8, we obtain
lim
an
=
a =>\J
n-+co
Α
no
\
a
n ~
a
\
<
l/ >
^Jf,
n>no
since l/m > 0 and l/m e 01. The rule of introducing the universal
quantifier (see formula (10), Section 8) may now be applied to this last
propositional function, which yields
(7)
lim an = a
n-* oo
=> Λ V Λ k . - * l <
m
n0
n>n0
]m
l-
10. EXAMPLES OF APPLICATIONS OF THE FUNCTIONAL CALCULUS
275
T o prove (5) we need only prove the implication converse to (7). T o do
so we use the following theorems concerned with real numbers:
(8)
for every real number there is a natural number which exceeds it,
(9)
for every real number ε > 0 and for every natural number
m,
if l/ε < m, then l/m < ε,
(10)
the relation "less t h a n " between real numbers is transitive.
By (8), the following proposition is true :
(11)
/W/l/e<m,
ε>0
BE St,me
Jf.
m
By (9), the following propositional function is true:
(12)
ε > 0=>
Λ 0/e < m=> l/m < ε).
m
On applying t o (12) the tautologies and rules (6), Section 6; formula (1),
Chapter XII, Section 8; formula (1), Section 8; formula (7), Section 2 ;
formula (7), Section 8, we obtain
(13)
/\\/\/e<m=>/\\/i/m<e.
e>0
ε>0
m
m
By detaching from (13) its antecedent (11) (under the rule of modus
ponens) we obtain
Λ V Um < e.
(14)
ε>0
m
By (10), the following propositional function is true:
(15)
(\ci„-a\
< l/m A l/m < ε) => \an-a\
< ε.
Since
\an — a\ < lfm A l/m < ε = l/m < ε A \an — a\ <
l/m
(see formula (32), Chapter XII, Section 12), the propositional function
(15) is equivalent t o
(16)
(l/m < ε A \an — a\ < l/m) => \an — a\ < ε.
By applying formula (16), Chapter XII, Section 12, to formula (16)
above we obtain
(17)
l/m < ε => (\an-a\
< l/m => \an-a\
< ε).
276
XUI.
FUNCTIONAL CALCULUS AND ITS APPLICATIONS
The rule of introducing the universal quantifier (see rule (10), Section 8) allows us to introduce the quantifiers
into the consequent
of (17), which yields
(18)
The law of splitting the universal quantifier over implication (see formula (5), Section 6) and rules (1) and (8), Section 8, yield the following true propositional function:
(19)
By applying the rule of hypothetical syllogism (see formula (1), Chapter
XII, Section 8), to formulas (18) and (19) above we obtain
(20)
The rule of generalization makes it possible to prefix propositional
function (20) with the quantifier!
, and rules (7) and (8), Section
8, make it possible to distribute them by prefixing both the antecedent
and the consequent of implication (20) with the quantifiers
this way we obtain
(21)
By detaching from (21) its antecedent in the form of (14) we obtain
Equivalence (6'), Section 5, makes it possible to introduce the quantifier
into the antecedent while changing into a universal one, and equivalence (7'), Section 5, makes it possible to introduce the quantifier
into the consequent. In this wav (22) vields
(23)
11. FORMALIZED MATHEMATICAL THEORIES
277
The consequent of (23) is equivalent to lim an = a. This and (23) yield
AVA,
m
n0
\<*n-a\
< l/m => liman
n>n0
= a.
n-*co
The proof of Theorem (5) is thus complete.
(VII) Theorem 5.3, Chapter VI (listed as (9)), i.e.
f(AnB)
czf(A)nf(B),
will now be proved.
The definition of the image of a set under a function (see formula (1),
Chapter VI, Section 5 ) ; formula (1), Chapter I, Section 3 ; and formulas
(32) and (33), Chapter XII, Section 12, yield
yef(AnB)
ΞΞ \J(xeAnBAy
= f(x)) = \/(xeAAxeBAy
= \J(xeAAy
= f(x) ΑΧ Ε B A y =
=
f(x))
/(*)).
X
On applying tautology (4), Section 6, we obtain
y ^f(A π Β) => \ / (χ Ε A Ay = /(χ))
X
A\J(xEBAy=
f(x)).
X
The consequent of the above implication is equivalent to y Ε f(A) A
Ay Ef(B). Hence
yEf(AnB)=>yEf(A)nf(B),
which proves the theorem. The remaining theorems in Chapter VI,
Section 5, can be proved similarly.
These examples of applications of the functional calculus in simple
proofs of theorems show how such proofs can be constructed by using
well-known tautologies and rules of inference; they also show that these
tautologies and rules are used by the mathematician even in the simplest
of arguments.
11. Note on formalized mathematical theories
The discovery of paradoxes in intuitive set theory (see Chapter I,
Section 10) shook the belief in the correctness of intuitive arguments
and resulted in the introduction of still greater precision in the con-
278
XIII. FUNCTIONAL CALCULUS AND ITS APPLICATIONS
struction of mathematical theories than was the case for axiomatic
theories. The so-called formalized mathematical theories represent this
more precise kind of axiomatic theories.
The first step in the formalization of a given mathematical theory
consists in the formalization of the language of that theory. Arguments
carried out in intuitive and axiomatic mathematical theories are formulated in every-day language. In formalized theories, everyday language
is replaced by a formalized language. T o do so we first list a system of
symbols, which are called the primitive signs of a given theory, and we
introduce rules which make it possible to construct in a precisely defined way, those expressions which we may use in a given theory. They
are called the well-formed formulas (or formulas) of the given theory.
Among the primitive signs of a theory we distinguish certain special
types. First of all, there are specific constants of a theory, that is, symbols
which denote all the primitive concepts of that axiomatic theory which
we formalize. For instance, when formalizing the axiomatic arithmetic
of natural numbers (see Chapter II, Section 1) we might adopt the
symbols: 1, + , · , = , < as specific constants. In every formalized
language there are also individual variables, that is symbols which denote any objects (individuals) with which a given theory is concerned.
For instance, in the formalized language of arithmetic we might adopt
the symbols xl9xl9
... to be the individual variables ; they would be
interpreted as symbols for any natural numbers.
Certain formalized theories may contain variables of higher types
i.e., symbols which denote any mathematical objects with which a given
theory is concerned but which are not individuals; they might be, for
instance, sets of individuals discussed in that theory, sets of sets, relations of m arguments holding between individuals, relations holding
between sets of individuals, etc. When formalizing the arithmetic of
natural numbers we might, for instance, introduce the variables
Xl9Xl9...
to be interpreted as symbols for any sets of natural numbers. If the formalized language of a theory does not contain variables of higher types,
then such a language is called elementary and a theory formalized by
means of an elementary language is called an elementary theory. It is
worth noting that the arithmetic of natural numbers can be formalized
so as to be an elementary theory.
11. FORMALIZED MATHEMATICAL THEORIES
279
The primitive signs include parentheses, which are used in the construction of well-formed formulas in order to ensure an unambiguous
reading of such formulas.
The primitive signs mentioned so far are used to construct atomic
formulas, which correspond to the simplest propositional functions that
occur in a given theory. F o r instance,
X\
' (*2+*3)
=
* 4 + ·*5>
Xl
<
^2+^3
are examples of atomic formulas in formalized arithmetic. Note, however, that not every finite sequence of primitive signs of a formalized
language is a (well-formed) formula, as it may be devoid of meaning,
as is the case, for instance of (xx = x2) = X3, which cannot be accepted
as a (well-formed) formula. The rules of construction of formulas make
it possible to construct exactly those formulas which are meaningful
(well-formed) from the primitive symbols in a purely mechanical way.
The symbols for propositional connectives, i.e., for conjunction, disjunction, negation, implication, and equivalence, and also the symbols
for quantifiers are added to the list of the primitive signs of a given
theory. In elementary theories, quantifiers bind individual variables
only. In those theories in which variables of higher types occur quantifiers may also bind those variables.
The atomic formulas enable one to form, by means of propositional
connectives and quantifiers, and with observance of the rules of construction of formulas, all the well-formed formulas of the formalized
language of a given theory. As has been said before, only formulas
of the formalized language of a given theory may be used in that theory;
this means that only strictly defined finite sequences of primitive symbols
of that language may be used.
A m o n g all the formulas of the language of a formalized theory we
single out those which are treated as analogues of the axioms of that
theory. They are called the specific axioms of a given formalized theory.
Obviously we assume that there is a method of verifying by means
of a finite number of steps whether a given formula is a specific axiom
or is not. F o r instance, when constructing a formalized arithmetic of natural numbers we may adopt the following formulas of its formalized
language as its specific axioms:
280
XIII. FUNCTIONAL CALCULUS AND ITS APPLICATIONS
0)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
(Π)
(12)
(13)
for every formula A in the formalized language of arithmetic. Axioms
(l)-(6) are axioms of equality, and Axioms (7)-(13) are a modification of the axiom system of Peano's arithmetic of natural numbers,
including the axiom of induction, which is represented in the form of
the schema of formulas (13).
The next stage in the formalization of an axiomatic theory consists
in the choice of logical axioms and rules of inference, which, taken
together, form the logical apparatus of that theory. Logical axioms are
selected from among those formulas which correspond to logical tautologies. Rules of inference are formal versions of some of the rules of
inference discussed in Chapters XII and XIII. It will be explained some-
11. FORMALIZED MATHEMATICAL THEORIES
281
what later how that selection of logical axioms and rules of inference
ought to be made.
N o w that we have at our disposal the formalized language of a given
formalized theory, the system of the specific axioms of that theory, the
system of logical axioms, and the system of rules of inference, we proceed to give precision to the intuitive concept of proof: its analogue in
a formalized theory is called a formal proof A formal proof of any
formula A in the language of a given theory is any finite sequence of
well-formed formulas of that theory such that every formula in that
sequence is either a specific axiom, or a logical axiom, or can be obtained
from formulas which occur earlier in the sequence by the application of
rules of inference, formula A being the last formula in the said sequence.
Those formulas for which there is a formal proof are called theorems of
a given formalized theory.
If in any formalized theory we suppress its specific axioms, we obtain
a formal logical system which is the logical foundation of that theory.
In such a logical system we can also carry out formal proofs by applying rules of inference to logical axioms or to formulas obtained earlier
from logical axioms by the application of rules of inference. Since logical
axioms are tautologies, and the rules of inference lead from tautologies
to tautologies, in such a logical system we obtain formal proofs only of
those formulas which are tautologies. The question naturally arises as
t o whether it is possible, in formalized theories, to select logical axioms
and rules of inference in such a way that in the resulting logical system
one can construct a proof of every tautologous formula. The answer to
this fundamental question is in the affirmative in the case of the elementary theories. This theorem, proved by K. Gödel in 1930, is one of the
most important theorems in mathematical logic. It is informative to note
that, for instance, it suffices to adopt as logical axioms all the formulas
in a given theory which are of the form *):
(a,)
<a 2)
*) Axioms (ai), (a 2), (a 3), (a 4) correspond, respectively, to the law of simplification, Frege's law, Duns Scotus' law, and Clavius' law Λ which were adopted, in
Chapter XTI, Section 13, as the axioms of the propositional calculus.
XIII. FUNCTIONAL CALCULUS AND ITS APPLICATIONS
282
( a 3)
(~
( a 4)
({~
A
=> (A
=*2?)),
A=> A)=>
A),
l/\(A~B(x))=>(A
( a 5)
*>/\B{x)%
X
X
where χ is any individual variable,
(/\Α(χ)^Α(τ)),
( a 6)
where A, B, C are any well-formed formulas of the formalized language of a given elementary theory. It is also assumed, in the case of
( a 5 ) , that A does not contain any free occurrence of x; in the case
of ( a 6 ) , τ stands for any individual variable or any individual constant
(if such do occur as primitive signs of the given theory, as is the case
for 1 in the formalized language of arithmetic), or for any expression in
the language of the given theory, constructed (in a manner precisely
defined) from individual variables or individual constants by means of
function signs, if such d o occur as primitive signs of the given theory
(as is the case for, e.g., (xt +1) · x2 in formalized arithmetic).
In such a system of logical axioms it suffices to adopt as the rules
of inference the rule modus ponens
A,
(A=>B)
Β
and the rule of generalization
Α
χ
(where χ is any individual variable).
Αχ)
Definitions (1), (2), (3), Chapter XII, Section 13 are also adopted,
as is the following definition of the existential quantifier:
\/A(x)
=
~/\~A(x)
X
(where χ is any individual variable).
Formal logical systems that have a formalized elementary language
and a system of logical axioms and rules of inference such that every
well-formed formula that is a tautology has a formal proof are called
first-order functional calculi.
11. FORMALIZED MATHEMATICAL THEORIES
283
A formalized theory is said to be consistent if no formal proof can
be carried out in that theory for a formula A and at the same time for
the formula ~ A.
The idea of formalism in mathematics, which resulted in the concept
of formalized theories, developed in connection with the Hilbert pro1
gramme ) , whose object was to construct a formalized theory that
would cover the whole of mathematics and t o prove its consistency by
employing the simplest of logical means. The Hilbert programme did
not yield the expected results. The following theorem, proved by K. G ö del (1931), resulted in its collapse. A formalized theory that would
cover the whole of mathematics would obviously contain the arithmetic
of natural numbers. Roughly speaking GödePs theorem mentioned
above states that a proof of the consistency of every formalized theory
that contains the arithmetic of natural numbers can be carried out only
in a mathematical theory which is more comprehensive than the one
whose consistency is to be proved. In particular, a proof of the consistency of formalized arithmetic can be carried out only in a mathematical theory which contains the whole of arithmetic and also other
theorems that do not belong to arithmetic.
In spite of the failure of the Hilbert programme, formalized theories
have retained their fundamental importance in the study of the philosophical foundations of mathematics. Because of their precise structure
formalized theories are an object of study that requires mathematical
precision and mathematical procedures. They may be treated as an object
of mathematical research. The mathematical theory concerned with the
2
).
study of formalized mathematical theories is called metamathematics
The most important problems with which metamathematics is concerned are : consistency, completeness, and decidability. Each of them will
now be discussed briefly.
GödePs above-mentioned result, concerning the proofs of the consistency of formalized mathematical theories has had a decisive effect
on research in that field by bringing about a change in the methods of
1
) David Hilbert, the eminent German mathematician, outlined his ideas in
1904. Serious work to put that programme into effect was started by Hilbert and
his co-workers P. Bernays, W. Ackermann, J. von Neumann and others only in 1920.
2
) The term metamathematics was introduced by Hilbert.
284
ΧΠΙ.
FUNCTIONAL CALCULUS AND ITS APPLICATIONS
proving consistency. Instead of striving for absolute (i.e., direct) proofs
of the consistency of formalized theories mathematicians have come to
confine themselves largely to relative proofs which consist in demonstrating that a theory under consideration is consistent if a certain other
formalized theory (e.g., the formalized theory of natural numbers) is
consistent. All those proofs are based on the profound conviction, even
though it cannot be proved, that the formalized theory of natural
numbers is free of inconsistency. This conviction is confirmed by the
experiences of mathematicians, accumulated throughout centuries.
A formalized theory is called complete if for every formula in the
language of that theory which represents a proposition, i.e. in which n o
free variables occur, there is a formal proof either of it or of its negation.
A formalized theory which does not have this property is called incomplete. Thus a theory is incomplete if there is an undecidable proposition in it, that is, a formula without free variables such that neither it
nor its negation is a theorem of the theory in question.
It might seem that a theory such as the arithmetic of natural numbers
should be capable of formalization in such a way as to make it a complete theory. But this is not the case in view of Gödel's remarkable
theorem (in the stronger version formulated by J. G. Rosser) stating that
every consistent formalized theory which contains the arithmetic of
natural numbers is incomplete. Gödel's theorem on the incompleteness
of arithmetic, proved in 1930 and published in 1931, is one of the greatest discoveries in the philosophical foundations of mathematics. It gave
rise to many important mathematical theorems, among them the theorem on the proofs of consistency, mentioned above, which were decisive in determining the directions of further researches in that field.
Finally, we should mention the problem of the decidability of formalized mathematical theories. A formalized theory is called decidable if
there is a method of determining, in a finite number of steps, whether any
given formula in that theory is its theorem or not. It is worth noting
that if a theory is decidable and if the decision algorithm is known,
then the study of the problems expressible in the language of such
a theory reduces to a purely mechanical procedure. In undecidable
theories the study of every problem requires an idea. It has turned out
that most mathematical theories are undecidable. The arithmetic of
EXERCISES
285
natural numbers is one of such undecidable theories. This result also
is due t o Gödel (1931). Those first-order functional calculi whose languages include binary relation symbols also are undecidable theories.
This theorem was proved by A. Church (1936).
Exercises
1. Indicate the free and the bound occurrences of the variables in the formulas
below :
2. In the formulas given below, eliminate the quantifiers whose ranges are restricted by propositional functions:
3. Symbolize the following propositions :
(a) The numbers 5 and 17 have no common divisor other than 1.
(b) There is a natural number such that no natural number is less than it.
(c) The system of equations: x+y = 2 , 2x+2y = 3 has no solutions.
4. Symbolize définitions of the following mathematical concepts:
(a) the limit of a sequence of real numbers,
(b) the greatest lower bound of a set X of real numbers,
(c) monotonicity of a function / : 0t->
(d) the Darboux property of a function f\0t-+
9t,
(e) differentiability of a function f\{x^0t\a<x<b}-+3fcm
any point of the
interval (a, b) and in the whole interval.
5. Symbolize:
(a) the condition of uniform continuity of a function f: {xe&: a < χ < b}-+ 31
in the interval (a, b),
(b) the condition of unboundedness of a sequence of numbers,
(c) Weierstrass' theorem on functions continuous in a closed interval,
(d) Rolle's theorem.
In cases (c) and (d) circle down the contrapositions.
6. Give examples of propositional functions φ(χ), ψ(χ), xeX,
following implications are false :
(a)
(b)
( V ?(*)=> V ψ(χ)) => Λ 0 w => VW) >
xeX
xeX
xeX
V (φ(χ) => ψ(χ)) => ( V φ(χ) => V
xeX
xeX
xeX
(y χ ) ) .
for which the
286
ΧΠΙ.
FUNCTIONAL CALCULUS AND ITS APPLICATIONS
7. Show that the following schemata represent functional tautologies:
(a)
(b)
Λ
^ v w ) => ( Λ <ρ( ) ο Λ v w ) >
χ
Λ 0 w ^ ν w ) => ( V y w o V ν w),
JCEA"
(c)
.ΧΕΑ"
(Λ
=> v w ) Λ Λ
=> Λ ν w ,
XEA"
A-EX
(d)
JCEA'
(Λ
VEA"
W => V W ) A V φ(χ)) => V V W ·
XEA"
ΧΕΑ"
XEA"
8. Prove De Morgan's laws, formulas (4) and (5), Section 4, for quantifiers with
ranges restricted by propositional functions.
9. Prove, by means of the functional calculus, the following theorems in the
algebra of sets :
(a)
Α-Β = OoA
a B,
(b)
{jAt-\jBtcz\J(At-Bt),
teT
(c)
teT
teT
(~)Α,-Γ)Β»
Γ)(Α,-Β,)<=
(d)
teT
teT
( J (AtnBt)cz
\jA,n
teT
teT
teT
B
U tteT
10. Show that the following schema is a functional tautology:
V
V ( V W A V W ) O ( V ?(*)Λ V
xeXyeY
xeX
vOO).
yeY
Use this tautology to deduce the following equation:
U U W M )
teT seS
=
U
teT
^
U
seS
^
CHAPTER XIV
l
E L E M E N T A R Y C O N C E P T S O F ABSTRACT ALGEBRAS )
1. Abstract algebras
m
Let A be a non-empty set. Every mapping o: A -> A, i.e., a function of m variables ranging over the set A and with values in A will be
called an m-argument operation on A. The case m = 0 is also admitted:
a zero-argument operation on A is interpreted as a constant element
ο e A.
If m = 1, i.e., if ο is a one-argument operation on A, we shall usually
write oa instead of o(a), a e A. If m = 2, i.e., if ο is a two-argument
operation on A, we shall usually write a1 ο a2 instead of o(ax, a2), αγ, a2
eA.
A subset A' c ^4 is closed under an m-argument operation ο on ^4 if
m
for every {ax, ..., am) e A
(1)
(a1eA'A
,
... A f l m 6 i 4 ) = ^ ö ( a 1 , . . . , f l m ) e i 4 ' .
Every system;
$1 = (A,oi9
on),
where Λ is a non-empty set and oj9 j = 1,
2
A ) , will be called an abstract
algebra,
is an operation on
or, briefly, an algebra.
For
every j = 1, . . . , n, m(j) will denote the number of the arguments of an
operation o}. If A has only one element, the algebra is called degenerate
or trivial.
0 An exposition of abstract algebra may be found in P. M. Cohn, Universal
algebra, New York 1965, or in G. Grätzer, Universal Algebra, D. Van Nostrand,
Princeton 1968. See also Moderne algebra by B. L. van der Waerden, 2 vols., Berlin 1930, 1931, and A survey of modern algebra by G. Birkhoff and S. Mac Lane,
New York 1953.
2
) The concept of abstract algebra as introduced here can be generalized by
admitting an infinite number of operations, and by allowing these operations to have
infinitely many argument places.
288
XIV.
CONCEPTS OF ABSTRACT ALGEBRA
Examples
(I) Let Ζ be the set of integers, + the operation of addition, and
the operation of multiplication. 3 = C3\ + , · ) is an abstract algebra.
(II) Let Jf be the set of natural numbers, + the operation of addition, and · the operation of multiplication. 9ΐ = (Jf 9 + , · ) is an
abstract algebra.
(III) Let Χ φ Ο be any set, and 2*, the family of all subsets of X.
X
$1 = (2 , u , n , — ), where u , n , — are, respectively, the set-theoretic operations of union, intersection, and complementation, is an abstract algebra. More generally, every field of sets (R, u , n , - ) (see
Chapter I, Section 7), where R is a non-empty family of certain subsets
of any set Χ φ Ο and is closed under the set-theoretic operations of
union, intersection and complementation, is an abstract algebra.
(IV) Let G be any subset of the set of all one-to-one mappings of
a set Χ φ Ο onto X; let G satisfy the following conditions: 1° the
\ï f g are in G, then gofeG,
3° if
identity transformation IxeG92°
1
f e G9 then f- eG. G is then a group of transformations (see Chapter
1
III, Section 4). <$ = (G9 ο , - , Ix) is an abstract algebra. It is said
briefly that every group of transformations is an abstract algebra.
(V) Let A = {0, 1 } . Three operations, defined by the tables given
below, namely ν , Λ , ~9 are introduced in A:
V
0
1
V
0
1
0
1
0
1
1
1
0
1
0
0
0
1
0
1
1
0
Note that the operations defined by the tables above coincide with
similar operations on the logical values 0 (falsehood) and 1 (truth)
(see Chapter XII, Section 2, formulas (6), (3), (9)). The abstract algebra
= ({0, 1}, ν , Λ , ^ ) is called the two-element Boolean algebra.
2. Subalgebras. Sets of generators
Let <2l = (A, ol9
on) be any algebra. A subset Α' Φ Ο of the set
A is called a subalgebra of the algebra ^ if A' is closed under all the
operations of the algebra ^1 (see Section 1). In particular, if Oj is a zero-
2. SUB ALGEBRAS. SETS OF GENERATORS
289
argument operation, then the condition o} e A' must be satisfied. Every
subalgebra of the algebra SI will be treated as an abstract algebra with
the operations on A restricted t o Ä (see Chapter III, Section 1).
Examples
(I) The algebra 9ί = (JV, + , · ) (Section 1, example (II)) is a subalgebra of the algebra 3 = (β, + , · ) (Section 1, Example (I)).
(II) The algebra ({Ο,X},
u , n , - ) is a subalgebra of the algebra
51 = (2*, u , n , - ) (see Section 1, Example (III)).
1
(III) Let ® = (G, ο , - , I#) be the group of all one-to-one transformations of the set 0t of real numbers onto 0t. Let G0 =
{/#,/},
where the function / i s defined by the formula f(x) = 1— x. G0 is then
a subalgebra of ©, since we have: I& ο Ix =
of = f ο I# = /,
f of = ΙΛ. The set G 0 is thus closed under the operation ο . Moreover,
1
a n
l
=
_ 1
^ i = Igt d f~
/ · The set G0 is thus closed under the operation .
Finally, I& e G0, which proves that G0 is closed under the zero-argument
operation on ©, namely the operation I#. We have also shown that
(<?ο> ο ,
Igt) is a group of transformations. It is a subalgebra of the
group ©.
2.1. The intersection of all subalgebras of a given algebra VI = (A,
..., on) is a subalgebra of SI.
ογ,
F o r let A0 c A be the intersection of all subalgebras of the algebra
SI. Assume that a1 e A0, ..., am{j) e Λ 0 · Those elements are in every
subalgebra of the algebra SI. Hence oj(a1, ..., a m ( J )) is in every subalgebra
am(j)) e A0 for every /
of the algebra SI, and consequently Oj(al9
= 1,
This proves that A0 is a subalgebra of SI.
It is proved in a similar way that
2.2. The intersection
a subalgebra of SI.
of any set of subalgebras
of an algebra
SI is
It follows from Theorem 2.2 that for every non-empty subset A0 of
the set A of the elements of an algebra SI there is a least subalgebra A'
of St that contains A0, namely the intersection of all those subalgebras
which contain A0. The set A0 Φ Ο, A0 cz A, is called a set of generators
of SI if the whole algebra SI is the least subalgebra that contains A0.
It is then said that A0 generates SI.
XIV.
290
CONCEPTS OF ABSTRACT ALGEBRA
Examples
(I) The set {1} generates the algebra 9ί = (Jf
Example (II)).
9
+ , · ) (see Section 1,
(II) The set { — 1} generates the algebra 3 = {β,
1, Example (I)).
+ , * ) (see Section
3. Similar algebras. Homomorphisms. Isomorphisms
on) and an algebra
We say that an abstract algebra $1 = (A, ol9
93 = (B9 o\, ..., o'k) are similar if η = k and if for every fixed j = 1, ...
..., η the operations o} and o) have the same number of arguments.
Examples
(I) The algebras 3 = {β, + , *) (see Section 1, Example (I)) and
9ΐ = (Jf 9 + , · ) (see Section 1, Example (II)) are similar.
(II) Any two fields of sets are similar.
(III) Any two groups of transformations are similar.
X
(IV) The field of sets 21 = (2 9
u, n, -)
and the two-element
Boolean algebra (see Section 1, Example (V)) are similar.
A mapping h: A -> Β that satisfies the following conditions:
(1)
h{oj{ax,
..., amU)))
= ο'^α^,
...,
h(a„U)))
for every j = 1, ..., η and for every sequence (al9 ..., am{j)) of elements
of the set A is called a homomorphism of an algebra $1 = (A, o A , ..., on)
into a similar algebra 93 = (B, o[, ..., o'n). The property of the mapping
h which is defined by conditions (1) is called the preservation of the
operations. Thus a homomorphism of an algebra into another preserves
the operations. A homomorphism of any algebra ^ = {Α9οΐ9
...,on)
into $1 is called an endomorphism. A homomorphism of an algebra
$1 = (A, ol9
on) onto a similar algebra 93 = (Β, o[, ..., o'n) is called
an epimorphism.
Examples
(I) Let SI =
+ ) , where & is the set of integers and + is the
operation of addition. Let 93 = ({1, — 1 , / , - / } , · ), where · is the
3. SIMILAR ALGEBRAS. HOMOMORPHISMS. ISOMORPHISMS
291
operation of multiplication. Let h: Ζ -> { 1 , — 1, /, — /} be a mapping
defined t h u s : for every integer k,
h(4k) = 1,
= ι,
h(4k+\)
h(4k + 2) = - 1 ,
h(4k + 3) =
-/.
The mapping A is a homomorphism of the algebra $1 onto the algebra
93, i.e., an epimorphism. T o prove this we must show that the equation
h(m
+ n) =
h(m)-h(n)
holds for any integers m and n. The easy proof is left to the reader.
(II) Let 21 = ( 2 ^ , u , n , - ) (see Section 1, Example (III)) and let
<2(0 = ({0, 1}, ν , Λ , ~ ) be the two-element Boolean algebra (see Section 1, Example (V)). A mapping h: 2^ -+ {0, 1} will be defined t h u s :
for every subset X cz Jf,
1
ifleX,
h
(
x
)
-
\0
ίϊΙφΧ.
The mapping A is a homomorphism of 21 onto 2 l 0 , i.e., an epimorphism.
T o prove this we must show that, for any subsets X and Y of the set
Jf of natural numbers,
(2)
h(XuY)
=
h(X)vh(Y),
(3)
h(XnY)
=
h(X)*h(Y),
(4)
h(-X)
= ~
h(X).
If h(XvY)
= 1, then l e l u 7 . Hence l e l o r l e F . Consequently,
either h(X) = 1 or h(Y) = 1. Hence h(X) ν h(Y) = 1. If h(XuY) = 0,
then 1 φΧυΥ.
Hence 1 φXand
1 φ Y, so that h(X) = 0 and h(Y) = 0.
Consequently h(X) ν h(Y) = 0 . Equation (2) thus holds.
If h(Xc\Y) = 1, then 1 eXnY.
Hence l e l a n d l e F , so that h(X)
= 1 and h(Y) = 1. Hence / Ζ ( Χ ) Λ Α ( 7 ) = 1. If h(Xr\Y) = 0, then
ΙφΧηΥ.
Hence 1 φX or 1 Y, so that /z(X) = 0 or h(Y) = 0 . Hence
h(X)Ah(Y)
= 0. This proves that equation (3) holds.
If h(-X)
= 1, then 1 e -X. Hence 1 φ X9 so that h(X) = 0 and consequently ~h(X)
= 1. If A(-Jr) = 0 , then 1 φ -X,
so that 1 e l .
Hence h(X) = 1 and ~ Α ( Χ ) = 0 . We have thus proved that equation
(4) holds. Since it is easy to show that h transforms 2^ onto {0, 1}, it
follows that A is a homomorphism of 21 o n t o 2 l 0 .
292
XIV.
CONCEPTS OF ABSTRACT ALGEBRA
(III) Consider the four-element algebra 21 = (A9 u , n , - ) , in which
1 } , and the operations u , n , — are defined by the
<2l = {09al9a29
following tables:
0
tfi
a2
1
0
ai
a2
1
η
0
«ι
a2
1
-
1
0
ai
ai
1
1
a2
1
1
1
1
1
0
0
0
0
0
0
ai
0
0
0
«2
öl
«2
0
ai
a2
1
0
ai
a2
1
1
a2
ai
0
1
tf2
1
Û2
1
Let A be a mapping of the set A into itself defined by
A(0) = 0,
Α(α 1) = 0 ,
A(l) = l .
h(a2)=\,
The mapping A is an endomorphism. The easy proof is left to the reader.
lA.Let
21 = (Α,οΐ9...,οΛ),
03 = (B9 o'l9...9 o'n) andQL = ( C , o ? , ...
..., ο*) be similar algebras. If A: A -> Β is a homomorphism of 21 into
93 and g: Β -> C is a homomorphism of 93 into (£, iAew /Ae composition
go his a homomorphism of 21 w/o <£.
The mapping g o A: ^4 -* C preserves the operations, for let al9 ...
···> amu) be any elements of A. Since A is a homomorphism,
(5)
Α(ο,.(α!, . . . , tf(m)7)) = o}(A(tf J , . . . , h(am(j))),
y = 1, . . . , η.
Since g is a homomorphism,
(6)
ifoXAfa), . . . , A(tf
))) = *J ( g ( A ( ö l) ) , ..., s(Afa. a ) )))
m(;)
. . . , g o h(amU)))9
= oj(goh(a1),
7 = 1 ,
But, using (5), we obtain
(7)
g ο A ( O , ( Ö ! , . . . , am(j)))
, ..., tfw(</))))
= gfaojfa
β
Α
β
Α
ί(οί( ( ι)»-» (^ω)))·
On comparing (6) and (7) we obtain
g ο hfofa,
. . . , Û F M 0) ))
=
o*(g ο A(UFJ), ..., g ο
h(amU)))
for 7 = 1, . . . , Λ , which proves that g o A is a homomorphism of 21
into <L
3.2. If the algebras 21 = (A, o1, ..., <?„)
93 = ( £ , o l , . . . , o'n) are
similar and if h is a homomorphism of 21 into 93, then the image h(A)
of the set A under A is a subalgebra of 93.
293
3. SIMILAR ALGEBRAS. HOMOMORPHISMS. ISOMORPHISMS
Obviously, h(A) czB.
then elements ai, ..., am{j)
Assume that bl9
of A such that
b1 = AOi),
(8)
···>
bm(j)
^(j) =
eh(A).
There are
h(am{j)).
This and (1) yield
o'j(bi9 . . . , 6 m ( i )) = o'jQiia^), ...9h(am(J)))
= h(oj(al9
.~9amU))).
This proves that
, ..., bm(j)) eh(A). The set h(A) cz Bis thus closed
under all the operations of 93, and hence it is a subalgebra of 93.
3.3. If a homomorphism h of an algebra SI = (A9ol9
. . . , o n ) into
a similar algebra 93 = (B9 o[, ..., o'n) transforms a set of generators of
SI onto a set of generators of 93, then h transforms SI onto 93.
If the assumptions made in Theorem 3.3 are satisfied, then the set
h(A) contains a set of generators of 93. By Theorem 3.2, h(A) is a subalgebra of 93. It follows from the definition of a set of generators and
from the remarks made above that h(A) = B.
3.4. Let A0 be a set of generators of an algebra SI = (A9 oi9 ..., on)9
and let h and g be homomorphisms of the algebra St into a similar algebra
93 =
(B,o[,...,o'n).If
h(a) = g(a)
for every
aeA'09
then
h = g,
that is9 h (a) = g(a),for
every a e A.
Let A' = {a e A: h(a) = g(a)}. The set A' is a subalgebra of SI, for
if α χ , ..., am(j) are in Ä9 then hfa) = g(a) for / = 1 , . . . , m(j). It follows
that
h(oj(al9
. . . , t f m ( J )) ) = o}(A(fli),
=
...9h(amU)))
...,g(flmu))) = g(oj(ai9
...9amU)))9
which proves that
·.., am{J)) e A'. Since A' is a subalgebra of SI
f
and contains the set A0 of generators of SI, hence A = A9 which proves
that h(a) =
for every ae A.
If a homomorphism f s a one-to-one mapping, then it is called a monomorphism. A mapping which is both an epimorphism and a m o n o morphism is called an isomorphism. Similar algebras SI and 93 for which
there is an isomorphism that maps St onto 93, are called isomorphic.
An isomorphism of any algebra SI onto St is called an automorphism.
294
XIV.
CONCEPTS OF ABSTRACT ALGEBRA
Examples
(I) Let SI = ({1, — 1 , / , — / } , * ). The following four mappings of
the set { 1 , — 1, /, —/} onto itself are defined t h u s :
(9) fi(x)
= 1 * x,
f-i(x)
= -1 '
fi(x) = i' x,
f-i(x)
= -i-
x.
Let 6 3 = ( { / i , / _ ! , / / , / - , · } , ° ) , where ο is the operation of composition of mappings. It can easily be verified that the set { / i
is closed under the composition of mappings, and hence under the
operation ο . We show that the algebras SI and 0 3 are isomorphic. Let
y, ζ be variables that range over the set {1, — 1, /, — / } . Note that
(10)
fy°fz
=fy.z-
This is true because, for every χ e { 1 , — 1, /, — / } , we have, under (9),
fy °fz(x)
= fy(fz(x))
=fy(z-x)=y(z'x)
= (yz)'X=
fy.z(x).
Let Λ be a mapping of the set of elements of SI into the set of elements
of © defined thus :
h(x) = fx
for every χ e {1, — 1, /, — / } .
It can easily be shown that Λ is a one-to-one mapping and that it carries
the set of elements of SI onto the set of elements of (S. Moreover,
under (10) we have
h(yz)
= fy.z
=f
f y y° f z
=
h{y)oh{z),
which proves that h is a homomorphism. Thus h is an isomorphism of
SI onto 0 3 , and hence these algebras are isomorphic.
(II) Let SI = ({0,X},
u , n , — ), where X is any non-empty set,
and Ο is the empty set. Also let 3ί 0 = ({0, 1}, ν , Λ , ~ ) be the twoelement Boolean algebra (see Section 1, Example (V)). These two algebras are isomorphic. For let h: {0,X}
-> {0, 1} be defined t h u s :
h{0) = 0, h(X) = 1. Obviously, h is a one-to-one mapping of
{Ο,Χ}
onto {0, 1} and the following formulas hold:
(11)
A(7uZ) =
h(Y)vh(Z),
(12)
h(YnZ)
h(Y)Ah(Z),
(13)
h(-Y)
=
= ~h(Y)
295
3. SIMILAR ALGEBRAS. HOMOMORPHISMS. ISOMORPHISMS
for every Y, Z e {Ο, X}. The above formulas result from Theorem 7.3,
Chapter I, and from the definitions of the operations ν , Λ , ~ in the
two-element Boolean algebra. They show that A is a homomorphism.
Hence A is an isomorphism of SI onto S l 0 .
(Ill) Consider an algebra SI = ( Q 0 , · ) , where Q 0 is the set of all
rational numbers other than 0, and the symbol · denotes the operation
of multiplication of rational numbers, restricted to the set Q0. F o r every
χ e Q o , let h(x) = I fx. It can easily be verified that A is a mapping of
the set Qo onto itself such that the condition: h(x-y) = h(x)-h{y)
is
satisfied. Thus A is an automorphism of the algebra SI.
3.5. Every algebra SI is isomorphic to itself. If A is an isomorphism
-1
of an algebra SI onto a similar algebra 93, then A is an isomorphism
of 93 onto SI. If h is an isomorphism of SI onto a similar algebra 93,
and if g is an isomorphism of 93 onto a similar algebra (£, then g o h
is an isomorphism of SI onto (£.
The identity mapping is an isomorphism of SI onto SI. Let A be an
isomorphism of SI = (A, ολ, ..., on) onto 93 = (B, o\, •••,o'n).
The
-1
mapping A is one-to-one and it maps 93 onto SI. Moreover, if bl9 ...
...,bmU)
are any elements of 93, then bt = Α(α Α), ...,bm{J)
=
h(am(j))
for some elements αΛ , ..., am^ of A. This and (1) yield
h-\o)(bi9
...,bmU)))
= h-^o^hia,),
1
= A" (h^jia,,
h(amU))))
..., am(J))))
= Ojfa
, ...,
amU))
1
=
Oj(h-\b1),...,h- (bm(j))),
-1
since A ο A = IA (see Chapter III, Theorem 3.2), and bx = Α(α,) if and
_1
only if ^ = A (^/) for / = 1,
m(j) (see Chapter III, Theorem 2.2,
-1
formula (5)). The last equation proves that A is a homomorphism of
-1
93 onto SI. Thus, A is an isomorphism of 93 onto SI. The last part
of Theorem 3.5 follows directly from Theorem 3.1 and Theorem 3.3,
Chapter III.
3.6. Let f be a one-to-one mapping of a set A0 of generators of an
algebra Si onto a set B0 of generators of a similar algebra 93. If there is
a homomorphism A of SI into 93 which is an extension of f and if there is
a homomorphism g of 93 into SI which is an extension of
then A is
-1
an isomorphism of SI onto 93 and A = g.
296
XIV.
CONCEPTS OF ABSTRACT ALGEBRA
It follows from the assumptions made in the above theorem and from
formula (8), Chapter III, Section 2, that
g o h(a) = g(h(a))
l
= t (f(a))
for every a e A0.
= a
This and Theorems 3.1 and 3.4 imply that
(14)
g o h(a) = a
for every element a of 21.
Likewise, the assumptions of the theorem and formula (9), Chapter III,
Section 2, imply that
1
h o g(b) = h(g(b)) = /(f-
for every b e B0.
(b)) = b
Hence, on applying Theorems 3.1 and 3.4, we obtain
(15)
ho g(b) = b
for every element b of 93.
It follows from (14) and (15) that h is a one-to-one mapping of the set
1
of the elements of 21 onto the set of the elements of 93, and that g = / r .
Consequently, h is an isomorphism of 21 onto 93.
The proof of the following theorem on the automorphisms of any
abstract algebra concludes the present section.
of
3.7. The set of all automorphisms
transformations.
of any abstract algebra is a group
Let H be the set of all automorphisms of any algebra 21. Note that
the identity transformation IA of the set A of all elements of 21 is an
automorphism of that algebra, and hence is in H. If hY e H and h2 e Hy
then, by Theorem 3.5, the transformation h2 ο h1 is an automorphism
of 21 and hence also is in H. Finally, if h e H9 then the converse trans1
formation / r is, by Theorem 3.5, an automorphism of 21 and thus also
is in H. This concludes the proof of Theorem 3.7.
Example
Consider an algebra 21 = (A, u , n , — ) , where A = {0,al9a2,
and the operations u , η , — are defined by the following tables :
u
0
at
a2
1
0
0
ai
a2
1
1
1
1
1
Oi
1
a2
1
1 a2
1 1
a2
1
0
0
0
0
1
al
Ol
a2
0
a2
a2
a2
Ol
«1
a2
1
1
0
0
0
1
0
0
0
0
0
1}
297
4. CONGRUENCES. QUOTIENT ALGEBRAS
The set of all automorphisms of this algebra consists of two transformations: the identity transformation I A and the transformation / defined
by the formulas:
/(0) = 0,
f(a1)
= a29
f(a2)
/ ( 1 ) = 1.
= al9
The easy proof stating that the set of transformations {IA>/}
of transformations is left to the reader as an exercise.
is a group
4. Congruences. Quotient algebras
Let 21 = (A, ol9
on) be any algebra. An equivalence relation Ä
on A is called a congruence on 21 if, for every operation
Oj,j=l,...,n,
the following condition is satisfied: for any elements a±, ..., am(j)9 b l 9 ...
···>
bm(j)
of
the
set
if
A,
a±
«
b l9
...,
a m () J
«
bm
(
,j )
then
Oj(ai,
« Ojibi,
..., am(J))
...,
b m
U ) ).
Examples
(I) Consider the field of all subsets of the set of natural numbers,
i.e., the algebra 21 = (2·^, u , n , — ) . F o r any subsets X and Y of Jf
we define
X* Yo(l
eXol
e Y).
The relation « thus defined is a congruence on 21. It can easily be
checked that it is an equivalence. Moreover,
if Xi » X2 and Yx « Y2, then
Z x u 7 i « Z 2 u 7 2 and X1 r\Yx « Z 2
n 7 2,
in view of the fact that 1 eΧγ \JYX Ο 1 eXx ν 1 e Y±. It follows from
the assumptions that
leX1oleX2
and
leY1oleY2.
Hence
1 eXx v l eY.ol
el2vl
eF2.
But
1 eJ2vl e 7 2 o l
e^uF^.
298
XIV.
CONCEPTS OF ABSTRACT ALGEBRA
Consequently,
l e ^ u ^ o l
eX2vY2,
which proves that X{ uYl κ X2 u 7 2 . It is proved similarly that X1 r\Y1
« X2nY2.
If X χ Y, then -X « - 7 , since then l e - I ^ l e - Y.
We have thus shown that « is a congruence on the algebra SI.
(II) Let 3 = («2Γ, + , · ), where ^ is the set of integers, and 4- and
• are the operations of addition and multiplication. Let ρ > 1 be any
fixed natural number. A relation « on 2£ is defined thus: for any integers x,y,
χ ~ y o p\x — ym
Thus the relation » holds between χ and >' if and only if the difference
x—y is divisible by p. Since p\x — x, we have χ » χ for every ^ e J . If
χ « >>, then p\x-}\
and hence
/>l-(*->')
=
y-x-
Consequently, y χ χ. 1ΐ χ & y and y » z, then
ν and/?[ν — z. Hence
p\(x-y)+(y-z)
=
x-z.
Thus, p\x — z, which proves that x « z. We have thus shown that
the relation « is an equivalence relation on
We now show that
if x1 » x 2 and yx & y2, then ^ +>Ί « x 2 +J2 and X i * ^ x 2 * ·
If the assumptions made above are satisfied, then p\Xi— x2 and
p\yi —yi - It follows that
=
(xi+yù-ixi+yi),
p\(xi-x2)+(yi-y2)
which means that xx +yx « x 2 +>> 2.
It also follows from the same assumptions that
ρ\(χι-χι)'yi
+ (yi-y2)'x2
= χι · yi-x2y2This proves that χλ · yx & x2- y2. We have thus shown that the relation
« is a congruence on the algebra 3 .
If « is a congruence relation on an algebra SI = (Α, ολ, ..., o„), then
we may form a set Α\κ, of all equivalence classes of the relation « on
Λ (see Chapter VII, Section 1). In accordance with the convention
adopted earlier, an equivalence class determined by any element a e A
s denoted by ||#||. By definition,
(1)
IMI =
{xeA:a*x}.
299
4. CONGRUENCES. QUOTIENT ALGEBRAS
By the principle of identification of equivalent elements (see Chapter
VII, Theorem 1.1), equivalence classes are non-empty and disjoint, and
their union equals A. Operations corresponding t o the operations oj9
j = 1
,
in 21 will now be defined on elements of the set A / » :
(2)
0*(||*ιΙΙ,
\\amU)\\)
= \\oj(al9
...,
tfm(i))||.
This definition is correct because, under the definition of a congruence
relation, the right-hand side of (2) does not depend on the choice of
the elements al9
am(j) from the equivalence classes
. · . , ||tf/nU)llFor if a x « bi9
am(j) « bm(j)9
then
..., amU))
Oj'di,
« oj(b1,
...,
bmU))
= \\oj(bl9
...,
bmU))\\.
and hence
...9amU))\\
\\Oj(al9
The resulting algebra
«/«
=
(Afn9o*l9...9o*)
is called the quotient algebra of the algebra 21 modulo κ . It follows
immediately from (2) and from the definition of homomorphism that
the mapping h : A -» A/& , defined by the formula: h{a) =
for every
a e A9 is a homomorphism of 21 onto 2 1 / « . It is called the natural
epimorphism of 21 onto 2 1 / « . A quotient algebra 2 1 / « w /Aw.s ÛAZ e/?/morphic image of the algebra 21.
Examples
jr
(Γ) Consider the algebra 21 = {2 9 u , n , — ) and the congruence
relation « on that algebra as analysed in example (I). Let Ρ be the set
of even numbers. Obviously, 1 φ Ρ, but 1 e —P. It follows from the
definition of the relation « that there are only two equivalence classes
of the relation « in 2 ^ :
||P|| =
\\-P\\
{
I
C
/
:
P
Ä
= {X czjr: -ρ
I
}
=
~ χ)
{X czzjr: 1
φχ}9
cjr
X}.
e
= {χ
:
1
Obviously,
= \\Χ\\ο\φΧ
and
||-P|| =
and
||^|| =
Hence
HOU = | | P | |
\\X\\oleX.
300
XIV.
CONCEPTS OF ABSTRACT ALGEBRA
The quotient algebra 2 1 / « = ( 2 ^ 7 « , u * , n * , - * ) = ( { | | 0 | | ,
\\JT\\}9
u * , n * , — *), in which the operations u * , n * , — * are defined by the
equations
I M | u * | | 7 | | = \\XuY\\9
I M | n * | | 7 | | = \\XnY\\9
-*|M|=
\\-X\\
for every X, Y c J ,
is an epimorphic image of the algebra 21. The mapping h: 2^ ->
1 1 ^ 1 1 } , defined by the formula A ( X ) = ||-ST|| for every I c / 5 is the
natural epimorphism of 21 onto 2 1 / « . It can easily be checked that the
algebra 2 1 / « is isomorphic to the two-element Boolean algebra 21 0
= ({0, 1}, ν , Λ , ~ ) (Section 1, Example (V)). The isomorphism is
defined by the formulas g(\) = \\JT\\ and g(0) =
(IT) Let 3 = C2\ + , · ) be the algebra discussed in example (II),
and let Ä be the congruence on that algebra as defined in the same
example. It follows from the definition of the relation « that the equivalence class ||ra|| = {x e S£\ p\m — x] for every m e Ζ. In other words,
a class \\m\\ contains those and only those integers which, when divided
by p9 leave the same remainder as does m. Thus, for instance, if ρ = 3 ,
then the class ||7|| contains the numbers: 1 , 4 , 7 , 1 0 , 1 3 , . . . and the
numbers —2, — 5 , — 8 , ... It follows that for a fixed ρ there are ρ equivalence classes of the relation « on 2£9 namely:
l|0||,l|l||,...,ll/>-l||.
Thus the set i2f/« consists of ρ equivalence classes. In the quotient
algebra 3 / « , operations which, in accordance with (2), correspond to
the operations + and · in 3 , are defined by the equations
ΙΜΙ + Ί Μ Ι = I I W l l ,
I M I ' * I L v l l = \\*-y\\
for every
x9ye&.
The quotient algebra 3 / « = C ^ f / ~ , + * , ·*) is an epimorphic image
of the algebra 3 . The mapping h: Ζ -*
h(m) = \\m\\ for every m e&9
, defined by the formula
is the natural epimorphism of 3 onto
3/*.
The concept of congruence on an abstract algebra is closely linked
with that of epimorphism. As previously stated, each congruence « on
an.algebra 21 determines the quotient algebra 2 1 / « and the natural
epimorphism of 21 onto 2 1 / Ä . On the other hand, the following theorem
holds:
5. PRODUCTS OF ALGEBRAS
301
4.1. Let h be an epimorphism of an algebra SI = (A, ol9
on) onto
a similar algebra 93 = (Β, o[, ..., o'n). Then the relation « on A, defined
by the formula
Ö! Ä a2 ο
A(fli) =
h(a2),
is a congruence on SI. The algebra 93 is then isomorphic to the quotient
algebra S l / « , as a mapping g: A/&
-» B, defined by the formula
S(IMI) =
/
ö r
every a Ε A ,
is an isomorphism of SI/ « onto 93.
It follows immediately from the definition of the relation « that it is
an equivalence relation on A. Assume that
#1
~
bi ,
m(j)
~
a
^M(J)-
Hence
..·,
= Kbi),
Ä ( a m ( 7 )) =
h(bmU)).
This yields
=
A ( o , ( a l 5 ...,amU)))
=
..., Α ( Λ » ω) )
OJ(A(*I),
...,h(bm(j)))
= h(oj(bl9
i
m(y)
)),
which proves that
OyOi, . . . , α , „ ω ) « Ö ; (O 1 5 . . . , 4 w ( i )) .
W e have thus proved that « is a congruence relation on SI. It follows
from the definition of the mapping g that g maps S I / « o n t o 93 and
that it is a one-to-one mapping. It remains to show that g is a h o m o morphism. On applying (2) and formula (1), Section 3, and referring
t o the definition of g we have
£(0?(lk*ill>
lk*«u)ID) =
g(\\oj(al9
a m ( i )) ) = o'jQiiaJ,
= h(oj(al9
=
...9h(am(J)))
f
o j(g(\M\)9...9g(\\am(j)\\)).
This proves that g is a homomorphism, and thus completes the proof
of Theorem 4.1.
5. Products of algebras
Let ( S l r ) i er = ((At9o\9
. . . , ö ^ ) ) i er be an indexed set of similar
algebras (see Chapter IV, Section 1), and let A = Ρ At (see Chapter VI,
teT
XIV.
302
CONCEPTS OF ABSTRACT ALGEBRA
Section 1). The elements of the set A are functions a defined on the
set T, such that, for every te Τ, a(t)eAt.
On setting a(t) = at for
every teT,
we may write a = (at)tET,
in accordance with the convention adopted in Chapter III, Section 1. The product A = Ρ At
teT
may be treated as an abstract algebra with the operations OjJ = 1,
defined t h u s : for every
= (am{j)t)teT
of functions
=
(au)teT,
= (o)(alt9
...,
a{
...,
am(j)
which are in A,
...9amlj))
Oj{ax,
system
= Oj((alt)teT,
(am(j)1
)tGT)
am(j)t))teT.
Roughly speaking an operation Oj is performed on the elements of the
set A by performing the operations o) in each algebra Si, on the elements alt,
am(j)t.
The algebra (Α, ολ,
on) thus defined is called
the product of the algebras SI, and is denoted by Ρ $1,. In particular
teT
the product of two similar algebras,
f
= (A2,o[ ,
...,o'n') is the algebra
S^ = (Ax, o[, ..., o'n) and
S ^ x S l ; , = (A1xA2,o1,
where the operations oj9 j = 1
ments (all9 a12)9 ... 9 (amU)l9
amU)2)
Oj((all9a12)9
a
...,on),
r e
defined t h u s : for any eleof the set
AlxA2,
· . · , (amU)l,
amU)2))
=
.-.9amU)1)9
(Ojian,
Sl 2
o'/(a12,
...9am{J)2)).
Examples
(I) For every neJf,
let Sl„ be the two-element Boolean algebra
(Section 1, Example (V)). The product of the algebras Sl„ is the algebra
SI = ( Ρ An, ν *, Λ *, ~ * ) , whose elements are sequences (an)neJ^ such
neJf
that, for every η eJf, an = 0 or an = 1. The operations in SI are defined
by the following equations:
( t f i n W v *(a2n)ne.r
(ßln)nejr
A^(a2n)ne^
= (alnv
=
a2n)n^r,
(alnAa2n)^9
(II) Let SI = (0t, + ) , where @ is the set of real numbers. The
product S i x SI =
+ ) , where # is the set of complex numbers,
and + is the operation of addition of complex numbers. This is true
303
6. ALGEBRAIC FUNCTIONS
because, by the definition of the product of algebras, for every pair
(JC, y) and (w, t) of real numbers we have
(χ, y) + (u, t) = (x + u,y + t),
and this equation defines the sum of complex numbers ζγ =
(x,y)
and z2 = (w, 0·
6. Algebraic functions
. . . , o n ) , the set 3 3 i of all algebraic
Given an algebra 21 = £A9ol9
So
functions on 21 is the least set of functions
v4 -> A which satisfies
the following conditions:
(fj) every function
defined by the equation
/ « ( * ) = */
0' = 1 , 2 , . . . ) ,
where χ = (xl9x2,
···)>
is in 3 s t ;
(f 2) if Oj (j = 1, . . . , « ) is a zero-argument operation on 21, then
the function f%9 defined by the equation
/«(*) = oj9
where χ = (xl9x29
...),
is in 3 ^ ;
(f 3) if Oj is an w(j)-argument operation on 21, m(j) > 0 , and if functions
... 5 /m(j)2i are in 3 s t , then the function f%, defined by the
formula
A W = Oj(f φ),
... ,/ -)«(x)),
mu
where χ =
, x2,
...),
is in 3 ^ .
The functions which are elements of the set 3 ^ are called algebraic functions on 21. It follows from the definition adopted above
that algebraic functions in fact depend only on a finite number of arguments but for technical reasons it is convenient to treat them as
0
functions defined on the product A* .
Examples
Let 21 = (.3?, + , ·, 0, 1), where & is the set of all integers, + is
the operation of addition, · is the operation of multiplication, and 0
and 1 are designated elements of 2£9 treated as zero-argument operations in
A function f){:
- • &9 defined by the formula:
= (Oi + l)- (*2 ' * 3 ) ) + ( * 2 + 0 ) ,
where χ = (xl9x29
···)>
304
XIV.
CONCEPTS OF ABSTRACT ALGEBRA
1
is an example of an algebraic function on SI. N o w let G5 = ( G , Ο, - , 1 j r )
be the group of all permutations of the set Jf of natural numbers.
A function f&: G*° - > G , defined by the formula:
1
χ
ι
/ © ( * ) = C * ! ° ιΥ
° 1JT>
where χ =
, x2, ...),
is an example of an algebraic function on ©.
Let S I = (A, ol9
on) and 93 = (Β, o[, ..., o'n) be similar algebras.
We shall say that algebraic functions f% : A*° -+ A and
: B*° Β
are corresponding algebraic functions on the algebras S I and 93 if fx
bears t o / © the relation ρ defined by induction as follows:
1° if the function fx is defined by the formula f%(x) = Xj and the
function fig is defined by the formula fs(x) = xj9 then Λιρ/©;
2° if the function / 3 l is defined by the formula fa(x) = Oj and the
function /© is defined by the formula fs(x) = o'j9 then / « ρ / © ;
3° if algebraic functions / l S (, ...,fmij)% on S I bear, respectively, the
relation ρ to algebraic functions / l S , . . . ,fm(J)sQ on 93, and if the function
f% is defined by the formula fn(x) = Oj(flfl(x)9
...9fm(J)% (x)) and if
the function/© is defined by the formula/© (x) = Oj(fm(x)9...
,fm(m(x)),
then fcxQfo.
Example
Consider the field of sets S I = (2·^, u , n , — ) and the algebra 93
= (ßf, + , · , — ) , where Z£ is the set of all integers, and + , · , — are,
respectively, the operation of addition, the operation of multiplication,
and the one-argument operation which with any number a associates
the number —a. The algebras S I and 93 are similar. The algebraic
f u n c t i o n s / ^ a n d / © , defined, respectively, by the formula:
U
/«(*)
= ((*i
/©(*)
= ( ( * i + * 2 ) '
* 2 ) <^(*2
(*2 +
u
-xx,
* 3 > ) + - * i ,
are examples of corresponding algebraic functions on SI and 93.
The definition of corresponding algebraic functions on two similar
algebras, as adopted above, can easily be generalized so as to embrace
1
any class K ) of similar algebras. This generalization is left t o the
reader as an exercise.
l
) The axiom system of set theory as adopted in Chapter I does not suffice for
the analyses carried out in Sections 6-9 of this chapter. These analyses require the
6. ALGEBRAIC FUNCTIONS
305
Some important properties of algebraic functions will now be discussed.
6.1. Let f% and fa0 be corresponding algebraic functions on an algebra
21 = (A, ol9
2 l 0 = (Αθ9 ox, ..., on).
on) and on its subalgebra
Then
the following equation holds:
(4)
=ΛΜ8».
fa
Q
In case 1 ° the theorem is self-evident. Likewise, in case 2°, since 21 0
is a subalgebra of 21, the set A0 is closed under every operation in 21;
in particular, every designated element of 21 is in A0 and accordingly
fa{x)
and
= oj
fa0(x)
=
oj9
which proves that formula (4) holds. Assume that
/«(*)
=
•••,/M(,)9(W),
fa»(x)
=
0 / ( / i » o( * ) >
/«<;)«<>(*)) ·
and that
/ L « O ( * )
=
/ L «
(*)MS°,
..· ,
f m U W o i x )
=
fm(m(x)/A$°
.
It follows from these assumptions that if χ e A§°9 then /f$t0(jc) = / i 2 i ( x )
for i = l , . . . , m ( j ) . It results from this and from the formulas given
above that (4) holds. This completes the proof of Theorem 6.1.
The next theorem refers to corresponding algebraic functions in any
algebra and in its epimorphic image.
6.2. Let h be an epimorphism of an algebra 21 = (A9 ox, ..., on) onto
a similar algebra 93 = (B, o[9
o'n)> Let fa and f% be corresponding
algebraic functions on 21 and 93, respectively. Then for every a = (al9
al9 ...) e ^4*0 the following equation holds:
(5)
h(Ma))
= fv(h(a)),
where h(a) = ( A ( f l l) , h(a2),
. . . ) 6 B»°.
Assume that the functions fa and fa are defined by the formulas
faix)
= Xj a n d / ö ( x ) = Xj. Then
h(fa(a))
=
h(aj)=fa(h(a))9
which proves that equation (5) holds.
use of an axiom system of set theory which includes a concept more general than
that of set, namely the concept of class (see Chapter I, Section 10 in fine, p. 25,
1
concerning the various axiom systems of set theory, and footnote ) on p. 119). If
such a concept is introduced, we can then use such concepts as the class of all sets,
the class of all cardinal numbers, the class of all similar algebras of fixed type, etc.
Such classes are not sets, but have certain properties in common with sets.
306
XIV.
CONCEPTS OF ABSTRACT ALGEBRA
Assume next that the function fa is defined by the formula fa(x)
= Oj. Since A is a homomorphism of 21 onto 93, hence h(Oj) = o). At
the same time assume that the function fa, being a corresponding
function of fa, is defined by the formula fa(x) = o). Thus we have
f
h(fa(a))
= h(oj) = o j
=fa(h(a)),
which proves that equation (5) holds.
It remains t o examine the case where the function fa is defined by
the formula
fa(x)
. . . , / M ( IS) ) ( ( x ) ) ,
= Oj(f1%(x),
the function fa is defined by the formula
fa(x) = o'j(fm(x),
. . . , / m ( i /©
) (x)),
and the functions fax and f^, i = 1, ...,m(j),
are corresponding algebraic functions on 21 and 93, respectively. T o prove the present case
assume that, for every i = 1, . . . , m(j) and for every a e A*°, the equation
(6)
h (fm
(a))
=fm(h{a))
holds. The definitions of the functions fa, fa, equation (6) and formula
(1), Section 3, yield
= h[oj(f1%{d),
~.,fm{m(a)))
= ο'^/ι(/^Λ(α)), ...,
= o'j[fm{h(ä)), . . . , / M O- ) £ ( A ( 0 ) ) )
h(fm(m(a)))
=fa(h(a)),
which proves equation (5) and completes the proof of Theorem 6.2.
T o conclude the present section we shall discuss algebraic functions
in a product of similar algebras (see Section 5).
Let ( 2 l f ) , e T = {{At,o\, . . . , ö i , ) ) i er be an indexed set of similar algebras. Let 21 = (Α, ox, ..., on) be the product of the algebras 21,,
teT.
Further, let fa and fat, t e T, be corresponding algebraic functions on the algebras 21 and 21,, t e T. Select any element α = (tti, a2, ···)
e A*°, where a, = (ait)teT, / = 1, 2 , ... Hence
(7)
a = ((alt)teT,
(a2t)teT,
···)·
x
The element a determines, for every t e T, an element a e Af°, which
is defined t h u s :
(8)
a* = (alt, a2t,
We prove the following theorem :
...).
6. ALGEBRAIC FUNCTIONS
307
6.3. Under the assumptions made above the following
(9)
Ma)
=
equation
holds:
(fat(a%er-
If the functions fa and fat, for t e Γ , are defined by the formulas
Mx) = Xj &ndfat(x) = xj9 then (7) and (8) yield
Ma)
= (ajt)ieT
and
fat(a<)
= ajt
for every t e T.
This yields (9).
If the functions fa a n d fat, for t e T, are defined by the formulas
Mx) = Oj and / o U (x) = o), then
/ a ( a ) = Oj
and
/ a , ( e 0 = o)
for ί G Γ.
Since ο), as zero-argument operations on the algebras 2l„ are designated
elements in those algebras, we have Oj = (ο))ίΕΤ by the definition of the
product of algebras. Hence we conclude that equation (9) also holds in
this case.
Assume now that the function fa is defined by the formula
x
M)
= ö/(/l3l(*)> ··· >/ma)2l(*))>
that the functions fat are defined by the formulas
fat(x)
=
«,(*), .·.,/«.(./)«,(*)),
and that the functions / i 9 i and fmt,
for every r e Τ and for every i = 1,
are corresponding algebraic functions
...,m(j),
/ eT,
on the algebras 21
and 21,, respectively. T o prove the present case we also assume that
(10)
fm(a)
= (fmt{a%zT
for every i = 1, . . . , m(j),
which states the validity of formula (9) for the functions fi% and f^t.
Under the definition of operations on a product of algebras (see Section
5) we obtain the following equation :
fa(a)
= Oj(fm(a),
= Oj({f1%t{d))tsT9
...,/™ ( ι /)«(α))
···. (/«(J)» f (^))w)
We have shown that formula (9) holds. The proof of Theorem 6.3 is
thus complete.
The theorems on algebraic functions, discussed in this section, will
be referred t o later.
308
XIV.
CONCEPTS OF ABSTRACT ALGEBRA
7 . Equationally definable classes of algebras
Consider a class Κ of all similar algebras of a fixed type (A, ol9 ...
...,on),
and f u n c t i o n s / of one variable, which associate an algebraic
function fa with each algebra 21 e Κ in such a way that all those algebraic functions fa which are values of one and the same function /
are corresponding algebraic functions (see Section 6).
Let K0 be a class of similar algebras, which is contained in K. It will
be said that the class K0 is equationally definable if there are two systems
of functions fS9 gS9 s G S, such that the following condition is satisfied:
(1)
an algebra 21 e K0 if and only if, for every s G S9 / S 3 J = gs<%.
The equation fs% = gs% holds if, for every a e A*°9 where A is the set
of the elements of the algebra 21, the condition fs%(a) = gs%(a) is satisfied.
The classes of equationally definable algebras play a very important
role in abstract algebra. Examples are given below.
Examples
(I) Semi-groups. Consider all similar algebras of the type {A, ο),
where ο is a two-argument operation. Let / be a function which with
each such algebra 21 associates an algebraic function fa, defined by the
formula fa(x) = (x1ox2)ox39
and let g be a function which with each
algebra 21 of this type associates an algebraic function g%9 defined by
the formula g<&(x) = x1o(x2ox3).
An algebra 21 is a semi-group if fa
= gs}[9 i.e., if and only if the following condition is identically satisfied:
(2)
(Χιθχ2)
ox3
=
xxo(x2ox3).
The class of all semi-groups is equationally definable.
The algebras specified below are examples of semi-groups: {Jf 9 +),
(Jf, · ) , where Jf is the set of all natural numbers, and + and · are
the operations of addition and multiplication; (G, o ) , where G is a
set of transformations closed under the operation of composition of
transformations, and ο is operation of composition of transformations;
X
X
X
(2 9 u ) , (2 , n ) , where 2 is the family of all subsets of a set Χ Φ 09
and u and η are the set-theoretic operations of union and intersection.
1
(II) Groups. Consider all similar algebras of the type (A9 ο , - , e)9
7. EQUATIONALLY DEFINABLE CLASSES O F ALGEBRAS
309
_ 1
where ο is a two-argument operation, , a one-argument operation,
and e9 a, zero-argument operation, i.e., a designated element of the set
A. Further, let/χ , / 2 , / 3 , gt, g2, g 3 be functions which with each algebra
g 2 Ä, g 3 i (
SI of this type associate algebraic functions fx%9 f2fi9 / 3 a ,
defined by the formulas
/!«(*) =
0
(*i
^2)
0
*3,
/ 2 a W = eoXl,
/ 3« ( x ) = * i
1 0
Χι,
Χ ι ο (x 2 ο χ 3 ) ,
gin(x)
=
g2*(x)
= Xl y
g3%(x) = e.
An algebra SI is a group if / i 3 = g i « , / 2 « = £ 2 2 1 , / 3 3 1 = £ 3 9 1 , that is, if
and only if the following system of equations is identically satisfied:
(Xl
Ο X 2) o
X3
= Χ1
ο
(χ2
ο X3),
e O XA =
XJ ,
X7
1
Ο XJ
= e.
The class of all groups is equationally definable. Each group of trans1
formations © = (G, ο , - , Ix) is an example of a group. The algebras
- 1
(2? 9 + , — , 0) and ( Q 0 , · , , 1), where 2ε is the set of all integers,
Qo, the set of all rational numbers other than 0, + and · are the
operations of addition and multiplication, — is the one-argument oper_ 1
ation which with every integer α associates a number — a, and
,
a one-argument operation which with every rational number α φ 0
s
- 1
associates a number I/a, are groups. An algebra Dt„ = (Mn9 ·, , /),
where Mn is the set of all non-singular matrices of order η with elements
in the set 0t of all real numbers, · is the operation of matrix multi_ 1
plication, , the operation which with every matrix associates an inverse
matrix, and / is a unit matrix, is a group. The algebra (£„ = (En9
+,
— , Ö), where En is the set of all vectors in an ^-dimensional space, 4- is
the operation of addition of vectors, — is a one-argument operation
which with every vector α associates the inverse vector — a, and θ is
the zero vector, is a group.
(Ill) Rings. Consider all similar algebras of the type (A , + , · , — , 0)9
where + and · are two-argument operations, — is a one-argument
operation, and Ο is a zero-argument operation. Further, let fi9
gi9
i = 1, . . . , 7, be functions which with every algebra of this type associate
algebraic f u n c t i o n s d e f i n e d by the formulas:
310
XIV.
CONCEPTS OF ABSTRACT ALGEBRA
/!*(*) = Χι ~\~ X ?
£m(*) — X 2
/mW
= ( x i + x 2) + * 3 >
#2«(*) =
= 0 +
£ 3 3 l ( * ) = Χι,
2
xl9
/4«(*)
= — Χι + X i ,
#4tl(*) =
/ K O )
= Xl
· x,
SsaC*)
FENIX)
= (*1 ' # 2 ) ' * 3 J
2
#72ΐ(*) =
fi%(X) — X l " ( ^ 2 ~T"*3) J
9
(x2+X3),
Xi +
0 ,
(Xi'X2)
+ (Xi '
An algebra SI is a ring if, for every i = 1, . . . , 7 , the e q u a t i o n / j g =
gi%
holds, i.e., if and only if the following system of equations is identically
satisfied :
=
X2~^~Xi9
=
Xl +
O + Xi =
xl9
Xi~\-x2
( * l + * 2) + * 3
— Χι +Xi
(X2+X3),
= O,
Xl ' X2 —— X2 ' Χ ι,
(Xl ' X2) ' x$ = Χι
Χι · (x2+X3)
{x2 ' X3),
= (Xi ' X2) + (Xi ' X3)'
9
The class of all rings is equationally definable. The algebra SI = ( J ,
+ , · , — , Ο), where
is the set of all integers and the operations are
ordinary arithmetical operations, is a ring. The algebra
= (P, + , *
—, 0 ) , where Ρ is the set of all polynomials in one variable with real
coefficients, + and · are the operations of addition and multiplication
of polynomials, — is the operation which with any polynomial a0 +
n
associates the polynomial — a 0 + ( — Qi)x+ ··· +
+ aix+ ... +anx
n
-+-( — an)x , and Θ is the zero polynomial, is a ring. A n algebra Θ
= ( S , + , · , θ ) , where S is the set of all sequences with rational terms,
and the operations in S are defined by the formulas:
(al9a29
...) + (bl9b29
.·.) = (a1+b1,a2
+ b2, ···)>
IFLl9a29
...) * (bl9b29
...) = (ar bi,a2
- b2,
-(al9a29
...) = (-al9
-a29
Θ = (0,0,...),
is a ring.
...),
...),
311
7. EQUATIONALLY DEFINABLE CLASSES OF ALGEBRAS
(IV) Lattices. Consider all similar algebras of the type (A, u , n ) ,
where u and η are two-argument operations. Let fi9 gi9 i = 1, . . . , 6,
be functions which with each algebra SI of this type associate
algebraic functions/jg, g i 3 I, defined by the formulas:
j
X± KJX 9
£ i 9 l ( * ) — ,X
= (Xi KJX ) \JX , g 2 « W = Λ Ί υ ( χ U X ) ,
Λ«(*) = (*! n x ) KJX , # 3 9 l ( * ) = x ,
/!«(*)
2
(3)
3
2
Λ»(*)
2
2
2
2
2
£4«(x) = x 2 η χ χ ,
n x 2,
= χλ
3
fs % (x) =
(^! nx2)
n x 3,
fe 51 (Χ) =
ΛΓα n ( x t
u x 2) ,
£5«(*)
x1 n(x2
=
£β«(*) =
nx3),
Χι-
An algebra SI is a lattice, if for every / = 1, . . . , 6 the e q u a t i o n / ) ^ =
holds, that is, if and only if the following system of equations is identically satisfied :
=
X± KJX2
X2
KJXX ,
(x u x ) KJX = jtj u ( x u x ) ,
(xj n x ) KJX = x ,
i
2
3
2
2
2
Xj nx2
= x2 ηχχ,
3
2
(4)
(*! n x 2 ) n x 3 = x A n(x2
xt
n(xj u x 2) =
nx3),
Xi.
The class of all lattices is equationally definable. The concept of lattice
is here introduced in a way other than that described in Chapter IX,
Section 4, but it can easily be proved that in a lattice as defined in
Chapter IX, Section 4, the operations u , η defined by the equations
aub = sup{tf,Z>} and ar\b = mS{a,b)
have the properties described
by the system of equations (4). On the other hand, it can also be easily
proved that in a lattice as defined in the present section the relation <
defined by the formula: a < b if and only if a nb = a, is an ordering
and that, under that ordering, avb = s u p { a , b} and anb = inf{tf, b).
Thus the set of elements of a lattice (in the sense defined above), ordered
by the relation < , is a lattice as defined in Chapter IX, Section 4. From
the practical point of view it is thus irrelevant which definition of a
lattice is adopted.
312
XIV.
CONCEPTS OF ABSTRACT ALGEBRA
A few more examples of lattices will now be given. A n algebra 91
= (R, u , n ) , where R is any family of sets closed under the operations of union and intersection of sets is a lattice. Such a lattice is
called a set lattice. A n algebra 21 = ( J ? ' , u , n ) , where 0t' is any nonempty subset of the set of real numbers, and the operations u and η
are defined by the formulas: aub = max(a,b),
anb = m i n ( a , & ) , is
a lattice. A n algebra 21 = ( F , u , n ) , where F i s the set of all functions
/: 0t -> ^ , which m a p the set ^ of real numbers into
and the
operations u , η are defined by the formulas: fug =
max(f,g),fng
= m i n ( / , g), is a lattice.
(V) Boolean algebras. Consider all similar algebras of the type
(21, u , n , — ) , where u and η are two-argument operations, and —
is a one-argument operation. Further, l e t / f , gi9 i = 1,
9, be functions which with every algebra 21 of this type associate algebraic funct i o n s / i 9 t, gm, of which Λ 3 ( , . . . , / 6 5 ί, gin,...,
g6% are defined by formulas
(3), and the remaining ones are defined by the formulas below:
fm(x)
= *i n ( x 2 u x 3 ) ,
g1%(x)
= (*i nx2)
fsvix)
= (xln-x1)KJx2,
g8<ii(x) =
x2,
fornix)
= (xlv-x1)nx2,
g9%(x)
x2.
=
u f o nx3),
An algebra 21 is a Boolean algebra if, for every / = 1, . . . , 9, the equation fw = gm holds, that is, if and only if system (4) of equations and
thr equations given below are identically satisfied:
xx n(x2
ux3)
= (*! nx2)
(xx n—x^uxz
=
x2,
(xj u — xl)nx2
=
x2.
u(xj
nx3),
The class of all Boolean algebras is equationally definable. Every
field of sets (R, u , n , — ) (see Chapter I, Section 7, and Section 1
above, Example (III)) is a Boolean algebra. The two-element Boolean
algebra (see Section 1, Example (III)) is a Boolean algebra.
The equationally definable classes of algebras referred to in examples
(I)-(V) are among the most important classes of algebras investigated
in mathematics, and each of them is an object of a mathematical theory.
The following theorem specifies the fundamental properties of equationally definable classes of algebras.
313
7. EQUATIONALLY DEFINABLE CLASSES OF ALGEBRAS
7.1. Let K0 be any equationally definable class of algebras. The following conditions are then satisfied:
(5)
ifVieKo
and 2 i 0 is a subalgebra of 21, then 2 l 0 G tf0,
if 21 G K0 and h is an epimorphism of 21 onto 93, then 93 G K0,
(6)
//algebras 21,, / G Γ , are in K0,
(7)
then their product is in KQ.
Assume that an algebra 21 is in the class K0 if and only if / Ä. 9t =
f or every s e S.
T o prove (5) assume that 21 G K0 and that 21 0 is a subalgebra of 21.
Let A0 be the set of all elements of 2 l 0 . The e q u a t i o n = gs% for every
s s S yields fy/Aty
=
for every seS.
This and Theorem 6.1
imply t h a t / s 3 Io = g s 3 io for every seS, which proves that 2 l 0 e K0.
T o prove (6) assume that 21 G !5Γ0 and that h is an epimorphism of
21 onto a similar algebra 93. Let b = (bt, b2, ...) be any sequence of
elements of 93. Since h is an epimorphism of 21 onto 93, for every bi9
i = 1,2,
there is an element at of 21 such that b> = A(a f ). Let
W e
trAUS
h a ev
a = (al9a29
...)·
6 = A(a).
(8)
This, the assumption that / s 9 l = g s 9( for every s e S, and Theorem 6.2
yield
/ , » ( * ) = / * ( * ( * ) ) = * ( / . « ( * ) ) = A ( f t „ ( e ) ) = &»(*(«)) =
for every 5 e S .
Since b is arbitrary, the equation fs^ =
holds
identically for every s e S, which proves that i> G K0.
It remains to prove (7). Assume that 21, G K0 for every r G T. It
follows that
(9)
fs%t
= g s S i,
for every
seS9teT.
Let 21 be the product of the algebras 21,, t e T, and let
(10)
α = ((tfirW, (a2t)teT,
...)
be any sequence of elements of 21. F o r every teT
ing sequence of elements of 21, :
(11)
ci = (ait,a2t,
consider the follow-
...).
It follows from (9) that
(12)
/ , „ , ( « ' ) = &„,(<!')
for every
seS9teT.
314
XIV.
CONCEPTS OF ABSTRACT ALGEBRA
Formulas (10), (11), (12) and Theorem 6.3 yield
/s«(a) =
(fs%t(<f))teT
= (gs<nt(a'))teT
= ^sa(a).
Since the sequence α is arbitrary, the above equation holds for every
sequence of elements of SI; we thus have fs% = g s il for every s e S,
which proves that the algebra SI is in K0. The proof of Theorem 7.1
is thus complete.
Properties (5), (6), (7) of equationally definable classes of algebras
are characteristic of those classes: it can be proved that if K0 is any
class of similar algebras and is such that conditions (5), (6), (7) are satisfied, then that class is equationally definable *). The proof is omitted
here in view of the rather elementary nature of this book.
8. Free algebras
Let Κ be any class of similar algebras. An algebra SI e Κ is called
a free algebra in the class Κ if it has a set A0 of generators such that
every mapping / : A0 -+ B, where Β is the set of the elements of any
algebra 03 e Κ can be extended to a homomorphism h: A -> B. The set
A0 is then called the set of free generators.
The following theorem will be proved :
8.1. If SI and 93 are two free algebras in a class K, such that the sets
A0 and B0 of free generators in those algebras are equipotent, then the
algebras SI and 03 are isomorphic.
Since A0 and B0 are equipotent, there is a one-to-one mapping
B0 of A0 onto B0. The assumption that SI is a free algebra
/: A0
implies that there is a homomorphism A of SI into 03 which is an extension of / . Likewise, the assumption that 03 is a free algebra implies
l
that the one-to-one mapping f~ : B0 -> A0 of B0 onto A0 can be extended to a homomorphism g of 03 into SI. On applying Theorem 3.6
we conclude that h is an isomorphism of SI onto 03.
Tt follows from the definition of a free algebra in a class Κ of algebras
]
) This theorem was proved by G. Birkhoff in On the structure of abstract
algebras, Proceedings of the Cambridge Philosophical Society 31 (1935).
8. FREE ALGEBRAS
and from Theorem 3.3 that every algebra 93 G Κ whose set B0 of generators is equipotent with the set A0 of free generators of a free algebra
SI is a homomorphic image of SI. Theorem 8.1 states that every two
free algebras in a class Κ and such that their sets of free generators are
of the same power are isomorphic.
Examples of free algebras conclude this section.
Examples
(I) Consider an algebra 3 = ißt, + , —, 0), where J f is the set of
all integers, + is the operation of addition, — is a one-argument operation which with every integer m associates a number — m, and 0 e J f .
The algebra 3 is a group (see Section 7, p. 309). The element 1 G J f
generates this algebra. It will be proved that 3 is a free algebra with
one free generator 1 in the class of all groups.
1
Let SI = (Α, ο , - , e) be any group. Hence the following system of
equations (see Section 7, p . 309) is satisfied:
(1)
{Χι ° X2) ο x3
= χ ι 0 (χ2
(2)
e ο xx =
(3)
1
XT
o Xl
χ3),
ο
xl9
.e
=
It follows from (1), (3), (2) that the following equations hold for
every element χ e A:
l
1
x~ ο (χ ο χ- )
= (χ
-1
1
χ) Ο χ-
= e ο χ-
1
1
Ο
1
1
=
χ- .
This and (3) yield
1
1
(χ- )-
1
1
ο χ-
1
1
1
ο ( χ - ο (χ ο χ - ) ) = (χ- )-
= e.
At the same time (1), (3) and (2) yield
1
1
1
( χ - ) " ο (χ-
1
1
ο (χ ο χ- ))
= ( ( χ - ) - ο χ- )
1
= e ο (χ ο χ- )
1
ο (χ ο χ- )
= χ ο
1
χ- .
It follows from the last two systems of equations that, for every element
x G A,
(4)
χ ο χ-
1
= e.
Now, (3), (1), (4) and (2) yield
1
χ ο e = χ ο (χ-
1
ο χ) = (χ ο χ- ) ox = eox
= x.
316
CONCEPTS OF ABSTRACT ALGEBRA
XIV.
We have thus proved that
(5)
for every x e A.
xo e = χ
It follows from (4) that, for every χ e A,
1
jrM*- )"
1
= e.
This and (2), (4), (1), and (5) yield
1
( x - i ) - i = e ο (χ- )= χ ο e =
1
1
(JC ο χ- )
=
1
ο (χ- )-
1
= χ ο (χ-
1
1
1
ο (χ- )- )
χ.
We have thus proved that
(6)
(χ
- 1
)
- 1
for every χ e A.
e
=
It follows from (2) in particular that
(7)
e ο e = e.
N o w (3) yields
l
ο e
l
ο e =
e~
=
and it follows from (5) that
e~
1
e" .
The last two equations yield
e-
(8)
1
= e.
T h e following equation which holds in 21 will also be proved:
1
( X i o x J - ^ ^ o x r
(9)
.
By (3), the following equation holds for every xl9x2
(xx ο x 2 )
- 1
© (χ1 ο χ 2 )
e A:
e.
=
This and (5), (4), (1) and (2) yield
1
_ 1
( * l ο Λ ^ ) " ο Χι = ((Χι ο χ 2 )
= ((Χχ ο Χ 2 ) -
=
1
° ΧΐΥ
1
ο χ 2) ο e
1
ο Χχ) ο ( χ 2 ο X J )
° (*1 ° *2>)
ο Χ2
1
Hence
(Χχ ο Χ 2 ) -
1
ο Xj =
1
XI .
= e ο Χ2
1
=
1
XJ .
317
8. FREE ALGEBRAS
This and (5), (4), (1) yield
(xl
l
ο X2y
1
ο X2)-
o e =
=
(Χί
=
((*ι ο * 2 )
- 1
( X j o X2y
° Χι) ° X I
1
l
ο (Λ-J Ο
= Χ2
1
1
XJ )
1
ο Χ]" ,
which proves (9).
The following notation will now be introduced: for every
xeA,
n
an element x x ο ... ο χ η 9 where xx = ... = xn = x , will be denoted by x
1
(n = 2 , 3 , . . . ) ; other conventions adopted a r e : x = x ; x° = e; for
- 1
every η = 1, 2 ,
an element ( x ) " is denoted by x~".
The following equations can be proved by reference to ( l ) - ( 9 ) :
(10)
x"
(11)
x
m+n
m
1
= (χ™)" ,
=
m
n
x ox ,
for every integer m and w. The proofs are left to the reader as exercises.
Let now / b e any mapping of the set {1} (the set of generators of
the algebra 8) into a set A of elements of a group 21. The m a p p i n g /
is extended to a mapping h: & -+ A t h u s : for every m e
it is assumed
that
(12)
Km) = ( / ( l ) ) * .
1
In particular, for m = 1 formula (12) yields A(l) = ( / ( l ) ) = / ( l ) ,
which proves that A is an extension of / . The mapping A is a homomorphism of .3 into 21, since the following equations hold by (12), (11)
and (10):
m+k
m
h'm + k) = (f(l))
= ( / ( l ) ) ο (/(l))
K-m)
= ((/(Ι))™)" =
= (/(I))-"
*(0) = ( / d ) )
0
1
fc
= him) ο h(k)9
(h(m))-\
= e.
This proves that 3 is a free algebra in the class of all groups, and that
the set {1} is its set of free generators.
(II) Let Τ be any non-empty set, and let Ε = {0, 1}. Consider the
T
product Q) = E , i.e., the set of all functions e: Τ -> Ε, which, under
the convention adopted in Chapter III, Section 1, will be denoted by
(et)teT.
318
XIV.
CONCEPTS OF ABSTRACT ALGEBRA
For every / e T, let 9)t be a subset of Q) consisting of all those functions e e Q) for which e(t) = et = 1. By definition,
(13)
et = 1}.
®t = {ee®:
Let D0 be the family of all sets 0 , , teT,
that is, D0 = ( Î / / W ,
and let D be the field of the subsets of Q), generated by D0.
Consider the algebra © = (Z), u , n , — ) . It follows from the
definition of this algebra that D0 is its set of generators. We will show
that © is a free algebra in the class of all fields of sets qua abstract
algebras, and that D0 is a set of free generators of ©.
Let 9v = (R, u , n , —) be any field of subsets of a set Χ φ Ο.
Further, let / : D0 -» R <be any mapping of D0 into R. For every / e T,
a mapping gt: X -• Ε will be defined thus:
( 1 4)
*<
W
=
I 1
if x e / ( 0 f ) ,
\0
i f x W ()
for every x e X. Assume also that
(15)
for every
g(x) = (gt(x))iET
xeX.
The mapping g maps X into ^ . Note that under (13), (14) and (15),
for every
teT,
(16)
g-*(%) = {xeX:
gt(x) = 1}
g(x) e &t} = {xeX:
= {xeX: x e / W } = /(^,).
Define
1
/?(Λ) = g' (A)
for every Λ e D.
It follows from (16) that h is an extension of the m a p p i n g / . Further,
from Theorem 5.10, Chapter VI, we obtain
x
1
l
h(A uB) = g~ {A uB) = g- (A)ug- (B)
=
h(A nB) = g-HA nB) = g-\A)ng~'(B)
=
h(-A)
= g->(-A)=
= X-h(A)
g-HD-A)
= -h(A)
=
for every
h(A)uh(B),
h(A)nh(B),
1
1
g- (D)-g- (A)
A,Be@,
which proves that h is a homomorphism of £ into 9Î. The theorem
under consideration is thus proved.
Note also that M. H. Stone's theorem (1936) states that every Boolean
9. CONSTRUCTION OF FREE ALGEBRAS
319
algebra is isomorphic to a field of sets regarded as an abstract algebra.
This and the theorem proved above show that the algebra φ (which,
being a field of sets, is a Boolean algebra) is a free algebra in the class
of all Boolean algebras.
9. Construction of free algebras for certain classes of algebras
Consider a class Κ of all similar algebras of a fixed type (A, ol, ...,
on).
Let U be any set of a power m and let V be an indexed set of symbols
and the parentheses
These symbols and the symbols ol9 ...,on
( , ) will be used for the construction of certain expressions which are
finite sequences of these symbols and are called terms of the algebras in the class K. We now give an inductive definition of the set
of all the terms of the algebras in the class K, constructed from the
symbols specified above. The set
is the least set of finite sequences
of symbols described above which satisfies the following conditions:
(%u)ueu-
(tj
for every u e U9
xue3T
( t 2)
( t 3)
if, for a fixed j (j = 1, ...,n) the y-th operation in any algebra
in the class Κ has zero arguments, then o} e,T,
if the y-th (y = 1, ...9n)
Κ has m(j)
mZT, then Oj(ri9
Now
ol9 ...9on
consider
operation in any algebra in the class
arguments, where m(j) > 0 and τΐ9
an
rmU))
abstract
rm(j)
are
eF.
algebra
$ = (^~, ox, ..., on)9
where
are operations i n ^ " defined t h u s :
a term Oj(ru
T w ( )i ) is the result of the operation Oj (j = 1,
n)
on the elements τί9
rm(j)
of&~9 if m(j) > 0; if the y-th operation
in any algebra in the class Κ is a zero-argument one, then Oj = Oj
is a zero-argument operation in the algebra
The algebra
called an algebra of terms, is obviously in K. It
can easily be shown that Κ is a set of generators of this algebra.
The following theorem holds:
9.1. The algebra £ = (3Γ', ox,
on) is a free algebra in the class
Κ of all similar algebras of the type (Α,ογ,
on). The set V is the
set of free generators in
320
XIV.
Every mapping / : V
algebra 21 = (A, o[9
CONCEPTS OF ABSTRACT ALGEBRA
A, where A is the set of all elements of any
o'n) in Κ can be extended to a mapping A:
-± A9 defined by induction with reference to the length of the terms
in the manner given below:
for every a M, u e U, let A(a M) = / ( a M ) ,
(1)
(2)
if the y-th (J = 1,
n) operation in any algebra in the class Κ
has zero arguments, then h(oj) = h{oj) = o'j9
(3)
if the y-th (j = 1, . . . , ri) operation in any algebra in the class Κ
has m(j) arguments, where m(j) > 0, and the elements A C O , ...
h(TmU)),
are already defined, then h(oj(r1,
..., r m a )) )
is
defined by the equations below:
h(oj(Tl9
T m ( )y ) ) = h(oj(Ti9
T m ( )y ) ) = ^ ( A C r O , . . . , A ( r m ( i )) ) .
The uniqueness of A is due to the fact that every term in Φ has exactly
one form, i.e., it is exactly one sequence of symbols. It follows from (1)
that A is an extension of the mapping / . It follows from (2) and (3) that
A is a homomorphism. This concludes the proof of Theorem 9.1.
Every mapping ν : V -> A of the set of free generators of the term
algebra ^ into the set A of all elements of any algebra 21 in Κ will
be called a valuation of the free generators of the algebra $ in the algebra 21. By Theorem 9.1, every valuation ν can be extended, by means
of (1), (2), (3), to a homomorphism of § into 21. That homomorphism
will be denoted by Vs% and will be called a homomorphism induced by
the valuation v.
The following theorem will be used later:
9.2. Let ν: V
Abe any valuation of the free generators of a term
algebra S in any algebra 21 = (A, o[, ..., o'„) which is in the class K,
and let τ = ( τ Α , τ 2 , ...) be any sequence of elements of
Then the
following equation
* « ( / S ( t ) )
(4)
where Vs}X(T) = ( ^ n f o ) , ν%(τ2)9
= / « ( 0 « ( t ) ) ,
. . . ) , holds for every pair fx,
responding algebraic functions on the algebras
fn
If the functions fx and / 9 t are defined by the formulas: fx(x)
=
Xi,
t r i ne
of cor-
§ and 21.
equation (4) holds, for we have
=
9. CONSTRUCTION OF FREE ALGEBRAS
Likewise, if the functions fa a n d / s l are defined by the formulas:
= oj9fa(x)
= ο}, then equation (4) holds, for we have
» a ( / i W ) = v9(pj)
321
fa(x)
= o] = A ( Ü Ä ( T ) ) .
N o w assume that the functions fa and fa are defined by the formulas :
(5)
faix)
= Oj(/iX(x)9
(6)
fa(x)
=
...,fmU)z(x))9
o'j(fin(x)9
where the functions fx and fan, for i = 1, ...9m(j)9
are corresponding algebraic functions on § and 21, respectively, and such that
(7)
«^(/isM) = / « ( « « ( τ ) )
for / = 1, ..., iwO").
Now (5), (6), and (7), and the fact that v% is a homomorphism yield
» a ( A W ) = * ψ Χ / ι ΐ ; ( τ ) , ···»/»u)3:W))
= O y ( / i a ( v a( T ) ) , . . . , / m ( J ) 9 l ( ^ W ) ) = / » ( ^ ( τ ) ) .
Equation (4) thus also holds in this case. This completes the proof
of Theorem 9.2.
Let K0 be any equationally definable class of algebras (at least one
of them non-trivial), contained on K. On a term algebra $ we introduce
a binary relation « defined t h u s : for every ordered pair ( r l 5 τ 2 ) of
elements of the set ZT9
(8)
T x « τ 2 if and only if for every valuation ν of the free generators of
the algebra $ in any algebra 21 which is in K09 the homomorphism v% induced by ν satisfies the condition
= ν%(τ2).
We will show that
9.3. The relation « on the set3T of all elements of the term algebra
defined by equivalence (8), is a congruence on
It follows immediately from the definition of the relation « that
« is an equivalence relation on the set &'. Assume that τ 1 , ...,
rmU)
and τ[9 ..., x'm{j) are any elements of the algebra § such that
(9)
r f « τ,'
for ι = 1, . . . , m ( y ) .
It is to be proved that if that is so, then OJ(T1 , . . . , r m ( i )) « ο ;(τΊ,..., T ^ ( J )) .
XIV.
322
CONCEPTS OF ABSTRACT ALGEBRA
Let ν be any valuation of the set of generators of the algebra $ in
any algebra SI = (A, o[, ..., o'„) in the class K0. It follows from assumption (9) and from the definition of the relation « that
Vsx(Ti) = vn{rl)
for i = 1, ..., m(j).
This and the fact that v% is a homomorphism of § into SI imply
that the equations given below hold:
τ
^ι(Λ·( ι> · · · >
r
«(j)))
=
v
r
°j( n( i)>
= O'J(V%(TÎ)9
···> ^stOmo)))
..., % ( τ ; ω ) ) = % ( ο , . ( τ 1 , ..., τ , ' „ ω ) ) .
This completes the proof of Theorem 9.3.
We shall now be concerned with the quotient algebra
which
will be denoted by S*. By the definition of the quotient algebra (see
Section 4), the elements of
are equivalence classes of the relation
« on the set^~, denoted by | | τ | | for each r e f . The set of all elements
of
will be denoted by F*. The operations o\,
o*(j) in
are
defined by the equations indicated in formula (2), Section 4.
We now prove the following theorem on algebraic functions on
the algebra
9.4. Let fa* be any algebraic function on the algebra §*, and let /V
be its corresponding algebraic function on the algebra
Then for every
sequence \\τ\\ = (\\Τι\\, | | τ 2 | | , ...) of elements of
the following
equation holds:
(10)
/*·(||τ||) =
||/Î(T)||,
where τ = ( τ 1 ? τ 2 , . . . ) .
As we see from the discussion in Section 4 (pp. 298-299), the mapping h:^~ -+&~*, defined by the formula: h(r) = | | τ | | for every τ
is an epimorphism of the algebra
onto the algebra § * . This and
Theorem 6.2 yield
/ϊ·(Ι|τ||) = / Ϊ · ( Α ( Τ ) ) = Α(Λ(τ)) = | Ι Α ( τ ) | | ,
which proves equation (10).
The theorem that follows is the goal of the analysis carried out in
the present section.
9.5. The algebra -£* is a free algebra in the class K0. The set of all
elements in the form | | a u | | for u e U, that is, the set of those equival-
9. CONSTRUCTION OF FREE ALGEBRAS
323
ence classes in ΖΓ whose representatives are free generators of the algebra % is the set of free generators of the algebra
If u Φ u\ then
α
αγχ
11 a j I ^ Ι Ι « ' I I ' ά hence the power of the set of free generators of the
algebra
equals the power of the set of free generators of the algebra
Assume that u Φ ü\ Let SI be any algebra in K0 such that the set of
its elements is of power greater than 1. Let a± and a2 be two different
elements of SI. Let υ be a valuation of the free generators of Φ in St
such that v((xu) = ax a n d v(oiu,) = a2. Then % ( α Μ ) = αΥΦ a2 = % ( α Μ , ) .
Hence | | a H| | Φ | | a M, | | .
We now show that
is in K0. Assume that the functions
fs,gs,
for each s e 5 , associate with each algebra Si in K0 algebraic functions
S U Cn a w a
t n at
e
K
fs*9g&
in
y
^
o if and only if the condition fs% = gs^
is satisfied for every s e S. Let
Ι|τ|| = ( l l r J U I I r . l l , . . . )
be any sequence of elements of
We must show that the equation
/ .(IIT|I) = Srt.(llt||)
(11)
ä
holds for every s e S.
T o achieve this consider any valuation ν of the free generators of
the algebra $ in any algebra SI e K0. Let v% be a homomorphism
induced by v. Then νη(τ) = ( ^ ( Ό , τ^(τ 2 ), ...) is a sequence of
elements of SI. Since SI e K0, the equation
(12)
=
gs^M)
holds for every s G S. It follows from Theorem 9.2 that
(13)
/ , « ( » » ( τ ) ) = ν « ( / 5 Ϊ( τ ) )
and
g s»(«>«(*)) =
v
«(#sî(*))
for every i e 5 . Formulas (12) and (13) yield
(14)
^C/k(*)) =
for every s e S.
This and the definition of the relation » on the set <T imply that
/ss(*) « ^ ΐ ( τ )
for every
seS.
As a result we have
(15)
IL/IsWII = | | Λ ΐ ( τ ) | |
for every se S.
324
XIV.
CONCEPTS OF ABSTRACT ALGEBRA
By Theorem 9.4, the equations
IL/kWH = / · ( | | τ | | )
Λ
and
\\gaX{x)\\
= * Λ. ( | | τ | | )
hold for every s e S.
It follows from these equations and from (15) that (11) holds. Since τ
is any sequence of elements of
we infer that equation (11) holds
for all τ, and thus we obtain fsX* = gsX* for every s e S, which proves
that the algebra
is in K0.
It follows immediately from the structure of
that the set of all
elements in the form ||a M|| for u e U generates § * .
Let / b e any mapping of the set of all elements | | a M| | , for u e U,
into a set A of all elements of any algebra 21 = (A,o[,
o'n) which
is in K0. It will be proved t h a t / c a n be extended to a mapping h: 3~*
-* A, which is a homomorphism of
into 21. The mapping / induces
a valuation ν of the free generators of § in 21, defined by
(16)
V(OL ) = / ( | | a | | )
u
U
for every
ueU.
By Theorem 9.1, the valuation ν can be extended to a homomorphism z\2[
of Φ into 21. Let τ and τ ' be any elements of S such that | | τ | | = | | τ ' | | .
Then τ » τ ' . It follows from Definition (8) of the relation « that
τ
^ 2 ΐ ( ) = *><a(O- Hence the condition | | τ | | = | | τ ' | | implies the condition
τ
τ
^ ι ( ) = ^ ι ( ' ) · Thus the equation
(17)
Λ(||τ||) = ^ ( τ )
defines the mapping h: 3Γ* -+ A. The mapping h is an extension of the
mapping / because, by (16) and (17),
A(||aJ|) = M « . ) = / ( l l « . l l ) ,
for every u e U. It can easily be shown that h is a homomorphism
of
into 21. Consider any elements | | ^ | | ,
| | T M ( |J| ) of the algebra
Ξ*. By formulas (2), (17), Section 4, and the definition of v<% we have
ACtfdlrJI, .··, Ι | τ „ ω | | ) ) = h(\\oj{TL9
r m a )) | | )
= V*(OJ(T19
r w ( i )) ) = ^ ( ^ ( T J ) , . · . ,
Vn(TMU)))
=
for every j = 1,
tion, then we obtain
^(11^11),
. . . , A ( | | r m ( i )| | ) )
In particular, if of is a zero-argument opera-
h(0f)
= VviOj) = Oj.
325
EXERCISES
We have thus proved that A is a h o m o m o r p h i s m of
completes the proof of the fact that
into % which
is a free algebra in the class
K0,
and so completes the proof of Theorem 9.5.
It follows from Theorem 9.5 that the equationally definable classes
of algebras (at least one of them non-trivial) have the important p r o p erty that, for every cardinal number m, there is in the class K0
algebra
a free
such that the power of the set of all its free generators equals m.
Exercises
1. Let 2t\ stand for the set of numbers 0, 1, ..., n— 1, and let © and 0 be operations on that set defined, respectively, as the remainders of dividing x+y and x-y
by n. Determine a set of generators of the algebra 3 n =
, θ » Θ) ·
2. Prove that the set {0, 3} is a subalgebra of the algebra 3 β = Α , Θ, Θ)·
=
a
3. Prove that the algebra 3 * i ^ o , Θ » Θ) Is homomorphic image of the
=
+ » '), where Ζ is the set of all integers, and + and · are the
algebra 3
arithmetical operations of addition and multiplication.
4. Determine all the automorphisms of the algebra 51 = (s&o),
= {a, b, c, e}, and the operation ο is defined by the table below:
ο
a
b
c
e
a
b
c
e
e
c
b
a
c
e
a
b
b
a
e
c
a
b
c
e
where $4
5. Prove that the following algebras are isomorphic: 51 = ißt> + , —, 0), where
0t is the set of all real numbers, 4- is the operation of addition, and — is the oneargument operation which with every real number χ associates the number — x;
+
1
f
and SB = ( ^ , ·, - , 1), where ^ is the set of all positive real numbers, · is the
_1
is a one-argument operation which with every
operation of multiplication, and
1
positive real number χ associates the number χ- .
6. Let Θ be the set of all congruences on any abstract algebra 51 = (Α, ox, ..., on).
Two operations, u and n , on the set Θ are defined as follows for any congruences Ä ι and « 2 which are in Θ and for any elements a, b of A :
if and only if there is a finite sequence a = x0f *i,
xn = b
fl(ÄiUÄ2)6
of elements of 51 such that, for every i = 1,
n, there is a y e {1, 2} such that
X i - i
~ j Xi j
x n ~ 2)b if and only if a ~ ι b and a ~2b.
326
XIV.
CONCEPTS OF ABSTRACT ALGEBRA
Prove that (0, u , n ) is a lattice in which a relation < defined by
£ ι < ~ 2 if and only if for every pair a9 b of elements of A the condition a
implies the condition a x2 b,
is the lattice ordering relation.
b
7. Consider the field of sets (2*, u , n , — ),
0 . A non-empty family ft
of subsets of X is called a /j/ter if the following conditions are satisfied: 1° if A e ft
and 5 e f t , then / i n ^ e f t ; 2° if / l e f t and B^X,
then , 4 u £ e f t . Prove that
X
the relation ν % on 2* which is determined in 2 by a fixed filter ft in the following
way:
<
A v% 5 if and only if
(-ΑνΒ)η(-ΒνΑ)β &9
is a congruence on the algebra (2*, u , n , — ). Determine the quotient algebra in
X
the case where the filter ft is prime, that is, where, for every A e 2 , exactly one
of the two sets, A and — A, is in ft.
1
8. Prove that every group 51 = (Α, ο, - , e») is isomorphic to a group of transformations (Cayley's Theorem).
Hint. As the required isomorphism use the mapping h which with every
element ae A associates the transformation fa: A-+ A defined by the formula
fa(.\) = ao x.
c
9. Let 51 = (A, u , n , — ) be any Boolean algebra. Any subset Ο # V Λ
which satisfies the conditions: 1° anbe V if and only if a e V and b e V; 2° Ö U Ö
e V if and only if a e V or 6 e V ; 3° for every aeA, exactly one of the elements
a, —a is in V, is called a prime filter of 31. It can be proved that the set ft of all prime
filters of any Boolean algebra is non-empty. Consider the field of sets (2^, u , η , — ).
Show that a mapping Λ: A-> 2^, defined by the formula
h(a) = { V e ft: ae V } ,
is a homomorphism of 51 into this field of sets. It can be proved that this homomorphism is an isomorphism (Stone's Theorem).
10. Let the algebra 51 = (A,oi9
o„) be a free algebra in a class Κ of similar
algebras, and let a = {a^, a2,.. ) be any sequence of distinct free generators of 51. Prove
that if, for any algebraic functions
and g^9 the condition f^(a) = g^iß) is satisfied,
= g<$ holds for the algebraic
then, in every algebra 33 which is in K9 the equation
functions
and g^ which correspond, respectively, to the functions / 9 ( and g^.
LIST O F I M P O R T A N T
G
1
/\
Φ 1
m\n
36
fix)
37
37
r
01 38
39
an 40
fe)
40
1 , # 2 , ...)
foi,...,a )
1
//x
gof
«
40
40
ft
42
43
45
48
50
51
tsT
00
U<"„
51
Al = 1
52
/er
c»
53
π=1
0,6)
60
JTxy 60
Λ: ρ y
62
63
63
ψ(χ, y), xeX,yeY
eW 68
^ + 70
®(é)
Ρ Λ
38
38, 235
xeX
Q
37
37
(fx)xeX
/ W
\/
39, 234
xeX
~
1, 181
Ο 2
{al9 ...,fl„} 2
C 2
D 2
=> 2, 183
<=> 2, 185
Φ 2
Φ 2
u
5
V
5, 180
Λ 5, 178
η 7
— 11
Α (.χ) 15
- Λ 15
Ψ(.Χ),ΧΕΧ
20
{* e
21
χ
23, 117
2
- 25
/ι 27, 100, 145, 153
η 28
A
SYMBOLS
AT
72
72
00
Ρ Λ
η=1
Α*
73
Λ«ο 73
73
64
LIST OF IMPORTANT SYMBOLS
328
m
Ρ Λ„
73
n= ι
AiX
1
Λ"
... X ^ m
73
74
ρ ( * ι , . . . , - v m)
76
...,*„,)
f(A)
78
_ 1
84
/ (0
Ä 90
78
IIJCII
A7«
91
91
x~y
99
F
loo
101
<ί7,6> 106
it ^ m 109
Ko
η < m
109
c 109
-< 123
sup A 132
inîA
132
~
141
ω 145
ω* 145
η US
λ 145
OL< β 156
α ^ β 157
α + 1 157
Ζ(α) 161
α 163
Z(m) 163
»(m)
163
Ω 174
174
1 27, 100, 145, 153, 178
0 100, 145, 152, 178
w(oc) = 1 178
w(a) = 0 178
h A 190
=
197
Ξ>
199
Κι
<&\- A
/ \ ç>(x)
xeX
224
234
\J<pM 235
f\<p(x)
v«
238
299
320
AUTHOR INDEX
Ackermann, W.
283
Bernays, P. 25, 283
Bernstein, F. 108, 110, 112
Birkhoff, G. 132, 287, 314
Bolzano, Β. 99
Boole, G. 6, 177
Borel, Ε. 110
Burali-Forti, C. 24, 162
Hauber, K. F. 203, 233
Hausdorff, F. 146
Hilbert, D. 187, 283
Kleene, S. C. 267
Kuratowski, K. 25, 60, 100, 116, 131,
132, 145, 149, 150, 172
Landau, E.
Cantor, G. 1, 23, 96, 99, 101, 107, 108,
110, 112, 116, 118, 138, 152
Cayley, A. 326
Chrysippus 177
Church, A. 285
Clavius 211, 212,224,281
Cohen, P. J. 170, 175
Cohn, P. M. 287
Dedekind, R. 121, 149, 150
DunsScotus 211,212,224,281
Euclid
24,212,214
Fraenkel, A. 25
Frege, G. 90, 177, 211, 212, 222, 224,
234, 281
Gödel, Κ. 25, 170, 175, 281, 283, 284,
285
Grätzer, G. 287
28, 94, 96
Lukasiewicz, J.
178, 187, 188
Mac Lane, S. 287
Moore, Ε. Η. 135
De Morgan, Α. 13, 17, 57, 58, 217, 246
Mostowski, Α. 25, 100, 116, 145
von Neumann, J.
25, 283
Origen 233
Peano, G. 1, 28, 68, 187, 280
Peirce, C S .
187,232,234
Post, E. 178
Rosser, J. B. 284
Russell, B. 24, 187, 234
330
AUTHOR INDEX
Schröder, Ε. 110, 134, 187, 234
Sheffer, Η. M. 188
Sierpinski, W. 1, 170
Stone, M. H. 326
Whitehead, A. N.
van der Waerden, B. L.
Zylinski, E.
287
234
Zermelo, E. 22, 25, 169, 170
Zorn, M. 131, 132, 172
188
SUBJECT INDEX
absorption laws 10
algebra, abstract 287
, Boolean 312
, degenerate (trivial) 287
,free 314
of terms 319
, quotient 299
, two-element Boolean 186, 288
algebraic functions 303
functions, corresponding 304
algebras, isomorphic 293
, product of 302
, similar 290
algebro-logical laws 217
rules 209
alternation of quantifiers, rule of
alternative negation 188
antecedent 60, 183
antisymmetric relation 67
apagogic proofs 213
ascending sequence of sets 59
associativity laws 6, 9, 217
asymmetric relation 66
atomic formulas 279
automorphism 293
axiom of choice 23
of difference 18
of existence 18
of extensionality of sets 18
of power set 23
of subsets 23
of union 18
axiomatic set theory 24, 25
theories 24
theories, formalized 278
axioms of algebra of sets 18
axioms of arithmetic of natural
numbers 27
of Boolean algebras 312
of first order functional calculi 281
of formalized elementary theories,
logical 281
of group theory 309
of lattice theory 311
of propositional calculus 224
of ring theory 310
of semi-group theory 308
of set theory 22
, specific 279
265
Bernoulli's inequality 36
binary propositional connectives 186
relations 61, 62
binomial theorem, Newton's 36
Birkhoff's theorem 314
Boolean algebra 312
algebra, two-element 186, 288
bound variable 235
Cantor-Bernstein theorem 110, 112
Cantor's method of constructing real
numbers 96
Cantor's theorem 107, 118
cardinal number 100
numbers, inequality of 108, 109
numbers, trichotomy theorem for 171
Cartesian product 60
product, generalized 72
Cayley's theorem 326
chain 130, 138
332
SUBJECT INDEX
characteristic function 89,116
choice, axiom of 23
, general principle of 170
Church's theorem 285
class 25, 119
of algebras, equationally definable 308
of a cut, lower 147
of a cut, upper 147
Clavius' law 224
rule 212
closed subset (under an m-argument operation) 287
system of implications 203
codomain of a function 38
of a relation 63
Cohen's theorems 170, 175
combination of k elements 34
commutativity laws 6, 9, 217
comparison of ordinal numbers 156,
157
complement of a set 15
complete formalized theory 284
completeness theorem 226, 230
complex function 40
composition of functions 45
conclusion in a rule of inference 194
condition, connectivity 138
, Moore-Smith 135
, necessary 202
, sufficient 202
connective, unary propositional 187
connectives, binary propositional 186
, extensional 187
, propositional 178
connectivity condition 138
congruence 297
conjunction 178
consequent 183
consistent formalized theory 283
set of formulas 226
constants, specific 278
constructing real numbers, Cantor's
method of 96
continuity, Dedekind's principle of 149
continuous linear ordering 147
order types 149
continuum hypothesis 116,175
, power of 109
contraposition, laws of 202, 216
contrary implication 203
converse implication 202
corresponding algebraic functions 304
cut of a linearly ordered set 147
, proper 147
decidable formalized theory 284
Dedekind's principle of continuity 149
theory of irrational numbers 150
deduction theorem 226
definition by transfinite induction, theorem on 167
degenerate (trivial) algebra 287
De Morgan laws 13, 17, 217, 246
laws, generalized 57, 58
dense linear ordering 146
order types 146
subset in a linearly ordered set 151
descending sequence of sets 59
detachment, rule of {modus ponens) 195
rules for equivalence 201
diagonal method 103
diagrams of ordered sets 122
dictum de omni 244
difference, axiom of 18
of sets 11
of sets, symmetric 25
directed set of elements 136
set of indices 135
directions 92
disjoint sets 10
disjunction 180
distributivity laws 10, 11, 55, 217, 255,
257
domain of a function 38
of a relation 62
SUBJECT INDEX
domain of a relation, j'-th 76
double negation law 217
Duns Scotus' law 224
rule 212
element 1
elementary formalized language 278
formalized theory 278
elimination and introduction laws for
quantifiers 249
empty set 1
endomorphism 290
enumerable sets 101
epimorphism 290
, natural 299
equal functions 39
equationally definable class of algebras
308
equipotent (equinumerous) sets 99
equivalence 185
classes of a relation 91
relation 90
equivalent propositional functions 198
propositions 197
excluded contradiction, law of 190
middle, law of 190
existence, axiom of 18
existential quantifier 235
theorems 248
exportation and importation law 216
extension of a function 39
extensional propositional connectives
187
extensionality of sets, axiom of 18
factorial 43
factors of a logical product (conjunction)
178
family of sets 19
of sets, indexed 50
Fibonacci's sequence 36
field of sets 19
333
filter 326
, prime 326
finite sequence of k elements 40
first element of linearly ordered set 140
first-order functional calculi 282
formal proof 224,281
formalization of a language 278
formalized language 223, 278
mathematical theories 278
theory, complete 284
theory, consistent 283
theory, decidable 284
theory, incomplete 284
theory, undecidable 284
formulas, atomic 279
, well-formed 223, 278
free algebra 314
generators, set of 314
variable 235
vectors 92
Frege's law 224
rules 212
function 37, 68
, characteristic 89, 116
, complex 40
establishing the equipotence of sets
99
, identity 43
, inverse 41
, Lejeune-Dirichlet 39
mapping (transforming) a set into a set
37
mapping a set onto a set 38
of m variables 78
, one-to-one 41
, real 40
restricted to a set 39
, value of 37, 68
functional calculi, first-order 282
calculus 234
laws 244
tautologies 244
functions, algebraic 303
334
SUBJECT INDEX
functions, characteristic 116
, composition of 45
, corresponding algebraic 304
, equal 39
gap 148
general principle of choice 170
generalization rule 263, 282
generalized De Morgan's laws 57, 58
generators, free, set of 314
, set of 289
Gödel's theorem 170,175,281, 283,284,
285
greatest element of a linearly ordered set
140
element of an ordered set 126
lower bound 132
group 309
, infinite symmetric 47
of transformations 47
, symmetric 47
Hauber's theorem 203
higher types, variables of 278
Hilbert programme 283
homomorphism 290
induced by valuation 320
hypothesis 202
, continuum 116, 175
hypothetical syllogism, rules of
idempotence laws 6, 9, 217
identical sets 3
identity function 43
image of a set under a function
implication 183
, contrary 203
, converse 202
, opposite 203
, simple 202
205
implications, closed system of 203
inclusion 2
incomplete formalized theory 284
indexed family of sets 50
indices, directed set of 135
individual variables 278
induction, principle of 27, 272
inductive proof 32
ineffective proofs 170
inequality, Bernoulli's 36
of cardinal numbers 108, 109
inference, rules of 194, 263, 280
infinite sequence 40
symmetric group 47
initial interval 153
intersection of sets 7
of sets, generalized 52
interval, initial 153
introduction and elimination laws for
quantifiers 249
intuitionism 178
inverse function 41
image of a set under a function 84
irreflexive relation 66
isolated ordinal numbers 166
isomorphic algebras 293
(similar) linearly ordered sets 141
isomorphism 293
joint negation
jump 147
188
Kuratowski-Zorn lemma
78
132, 172
language, elementary formalized 278
, formalization of 278
, formalized 223, 278
last element of linearly oredered set 140
lattice 133,311
, set 312
SUBJECT INDEX
law of associativity for the intersection
operation on sets 9
of associativity for the union operation
on sets 6
, Clavius' 224
of commutativity for the intersection
operation on sets 9
of commutativity for the union operation on sets 6
of distributivity of conjunction over
disjunction 217
of distributivity of disjunction over
conjunction 217
of distributivity of intersection over
union 10, 55
of distributivity of the existential quantifier over disjunction 257
of distributivity of the universal quantifier over conjunction 255
of distributivity of union over intersection 11,55
of double negation 217
, Duns Scotus' 224
of excluded contradiction 190
of excluded middle 190
of exportation and importation 216
, Frege's 224
of idempotence for the intersection
operation on sets 9
of idempotence for the union operation
on sets 6
of identity for equivalence 190
of identity for implication 190
of negation of equivalence 217
of negation of implication 217
, Peirce's 232
of simplification 224
of splitting the universal quantifier
into existential quantifiers 260
of splitting the universal quantifier
over implication 258
laws, algebro-logical 217
, De Morgan 13, 17, 217, 246
335
laws, functional
244
, generalized De Morgan 57, 58
governing the alternations of quantifiers
261
of absorption 10
of associativity in the propositional
calculus 217
of commutativity in the propositional
calculus 217
of contraposition 202,216
of idempotence in the propositional
calculus 217
on the relettering of quantifiers 260
, propositional 190
least element of a linearly ordered set
140
element of an ordered set 128
upper bound 132
Lejeune-Dirichlet function 39
lemma, Kuratowski-Zorn 132, 172
limit ordinal numbers 166
linear ordering 138
ordering, continuous 147
ordering, dense 146
linearly ordered set 138
logical product 178
sum 180
lower bound 131
bound greatest 132
class of a cut 147
many-valued logics 178
mapping of a set into a set 37
m-argument operation 287
ra-ary relations 75
mathematical induction, principle of 27
theories, formalized 278
maximal element of an ordered set 125
set of formulas 227
metamathematics 283
method, diagonal 103
336
SUBJECT INDEX
method of constructing real numbers,
Cantor's 96
of identification of equivalent elements
93
, truth-table 191
minimal element of an ordered set 127
minimum principle 30
modus ponens 195, 224
modus tollendo ponens 211
monomorphism 293
Moore-Smith condition 135
natural epimorphism 299
numbers 27
necessary condition 202
negation 182
, alternative 188
Joint 188
Newton's binomial theorem
symbols 33
non-enumerable sets 106
numbers, cardinal 100
, isolated ordinal 166
, limit ordinal 166
, natural 27
, ordinal 152
36
one-to-one function 41
sequence 165
operation, m-argument 287
operations, preservation of 290
opposite implication 203
opposition, square of 203
order type 145
types, continuous 149
types, dense 146
ordered pair 60
set 121
set, linearly 138
ordering, continuous linear 147
, dense linear 146
ordering relation 121, 138
relation, partial 121, 138
ordinal number, power of 163
numbers 152
numbers, comparison of 156, 157
numbers, isolated 166
numbers, limit 166
numbers, trichotomy theorem for 159
pair, ordered 60
paradox, Russell's 24
partial ordering relation 121, 138
Peirce's law 232
permutation of a set 43
power of a set 100
of the continuum 109
of an ordinal number 163
set 23, 117
set axiom 23
predecessor 150
predicate 20, 64, 76
premisses in a rule of inference 194
preservation of operations 290
prime filter 326
primitive concepts of an axiomatic theory 24
signs 278
principle, minimum 30
of choice, general 170
of continuity, Dedekind's 149
of identification of equivalent elements
92
of mathematical induction 27
product, Cartesian 60
, Cartesian generalized 72
, logical 178
of algebras 302
, set-theoretic 7
proof, formal 224, 281
, inductive 32
proofs, apagogic 213
, ineffective 170
SUBJECT INDEX
proper cut 147
subset 2
proposition, undecidable 284
propositional calculus 177
connectives 178
, binary 186
, unary 187
function of m variables 76
function of one variable 20
function of two variables 64
functions, equivalent 198
laws 190
tautologies 190
propositions, equivalent 197
quantifier, existential 235
, scope of 240
, universal 235
quantifiers with restricted range
quasi-ordered set 134
quasi-ordering relations 133
quotient algebra 299
range of a variable 20
real function 40
reductio ad absurdum 213
reflexive relation 66
relation, antisymmetric 67
, asymmetric 66
, binary 61, 62
, codomain of 63
, domain of 62
, equivalence 90
, irreflexive 66
, i-th domain of 76
, linear ordering 138
, m-ary 75
, ordering 121, 138
, partial ordering 121,138
, reflexive 66
, quasi-ordering 133
238
337
relation, restricted 129
, singular (unary) 15
, symmetric 66
, transitive 67
relettering laws for quantifiers 260
restricted relation 129
ring 310
rule, Clavius' 212
, Duns Scotus' 212
of alternation of quantifiers 265
of contracting the universal quantifier
over disjunction 264
of detachment (modus ponens) 195
of eliminating the existential quantifier 266
of eliminating the universal quantifier
266
of generalization 263, 282
of inference 194, 263, 280
of introducing the existential quantifier 265
of introducing the universal quantifier 265
of splitting the existential quantifier
over conjunction 264
, simplification 211
rules, algebro-logical 209
, Frege's 212
of detachment for equivalence 201
of hypothetical syllogism 205
Russell's paradox 24
satisfiability of a propositional function
21, 65, 77, 236, 242
Schröder-Bernstein theorem 110
scope of a quantifier 240
semi-group 308
sequence 40
, Fibonacci's 36
, finite, of k elements 40
, infinite 40
, one-to-one 165
338
SUBJECT INDEX
sequence of sets, ascending 59
of type a, transfinite 164
set 1
, empty 1
, enumerable 101
lattice 312
, linearly ordered 138
, non-enumerable 106
of arguments of a function 38
of elements of a space, directed 136
of formulas, consistent 226
of formulas, maximal 227
of free generators 314
of generators 289
of indices, directed 135
, ordered 121
, power 23,117
, quasi-ordered 134
theory 1
, well-ordered 152
set-theoietic product 7
sum 5
sets, ascending sequence of 59
, descending sequence of 59
, disjoint 10
, equipotent (equinumerous) 99
, identical 3
, indexed family of 50
, intersection of 7, 52
, union of 5, 51
signs, primitive 278
similar algebras 290
(isomoiphic) linearly ordered sets 141
simple implication 202
simplification law 224
rule 211
singulary (unary) relations 15
space (universe) 14
specific axioms 279
constants 278
square of opposition 203
Stone's theorem 326
subalgebra 288
subset 2
, closed under an m-argumerit operation 287
, proper 2
subsets, axiom of 23
successor 27, 60, 150, 153
sufficient condition 202
sum, logical 180
, set-theoretic 5
summands of a logical sum (disjunction)
180
symbols, Newton's 33
symmetric difference of sets 25
group 47
group, infinite 47
relation 66
system of implications, closed 203
tautologies 190, 244
terms, algebra of 319
theorem, Cantor-Bernstein 110,112
, Cantors' 107,118
, completeness 226, 230
, deduction 226
for cardinal numbers, trichotomy 171
for ordinal numbers, trichotomy 159
of Zermelo, well-ordering 171
on definition by transfinite induction
167
on transfinite induction 164
, Schröder-Bernstein 110
theorems, existential 248
of formalized theory 281
theories, formalized mathematical 278
theory, complete formalized 284
, consistent formalized 283
, decidable formalized 284
, elementary formalized 278
, incomplete formalized 284
, undecidable formalized 284
of irrational numbers, Dedekind's
150
SUBJECT INDEX
thesis 202
transfinite induction, theorem on 164
sequence of type α 164
transitive relation 67
transformation of a set into a set 37
transformations, group of 47
trichotomy theorem for cardinal numbers 171
theorem for ordinal numbers 159
trivial (degenerate) algebra 287
true propositional function 193
truth-table method 191
two-element Boolean algebra 186, 288
type, order 145
types, continuous order 149
, dense order 146
universal quantifier 235
universe (space) 14
upper bound 131
bound, least 132
class of a cut 147
unary propositional connective 187
(singulary) relations 15
undecidable formalized theory 284
proposition 284
union, axiom of 18
of sets 5
of sets, generalized 51
well-formed formulas 223, 278
well-ordered set 152
well-ordering 152
theorem of Zermelo 171
339
valuation of free generators 320
of propositional variables 230
value of a function 37, 68
variable, bound 235
,free 235
variables, individual 278
of higher types 278
vectors, free 92
Zermelo, well-ordering theorem of
171