MATH 705
Set Theory and Logic
Module 5
S
ETS
Objectives
After studying this module, you should be able to:
1. define set and its properties;
2. describe set by using either or both methods of naming sets;
3. enumerate, define, and give examples of the different kinds and relationships of
sets;
4. consider sets as elements of another set;
5. perform set operations;
6. use Venn diagram to illustrate the relationships and operations between sets;
7. consider sets as elements of another set; and
8. state the axiomatic development of set theory.
Introduction
The concept of set appears in all branches of mathematics. This concept
formalizes the idea of grouping objects together and viewing them as a single entity.
This chapter introduces this notion of a set and its members. Furthermore, it builds all
the basic properties, axioms, theorems, basic operations, and its applications.
Logic is formally discussed in the previous chapters, we indicate here the close
relationship between set theory and logic by showing how Venn diagrams, pictures of
sets, can be used to determine the validity of certain arguments.
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Set and Element
A set is a well-defined collection of distinct objects and is denoted by an upper
case letter.
An object that belongs to a set is called an element or member of the set and is
usually denoted by a lower case letter. The symbol “” denotes membership while “”
denotes non membership to a set. Thus, x  A, read as “x is an element of A”, means
that x can be found on set A while x  A, read as “x is not an element of A”, means that
x does not belong to set A.
Examples:
(1) Set of integers greater than 10.
(2) Set of vowels in the Filipino alphabet.
(3) Set of the colors of the rainbow
(4) Set of numbers that satisfy the equation x2 – 9 = 0.
(5) Set of colleges in CLSU.
Ways of Describing Sets
1. Roster/Tabular Method is a method in which the elements in
the
given
set
are
listed or enumerated, separated by a comma, inside a pair of braces.
2. Rule/Descriptive Method is a method in which the common characteristic of the
elements are defined. This method uses set builder notation where x is used to
represent any element of the given set.
Examples:
(1)The distinct letters in the word “algebra”
Let A be the set of distinct letters of the word “algebra”.
Roster form: A = { a, l, g, e, b, r }
Rule form:
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A = { xx is the distinct letter of the word “algebra” }
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(2) The colors of the rainbow
Let B be the set of the colors of the rainbow.
Roster form: B = { red, orange, yellow, green, blue, indigo, violet }
Rule form:
B = { xx is the color of the rainbow }
(3) An even prime number
Let C be the set of even prime number
Roster form: C = { 2 }
Rule form:
C = { xx is an even prime number }
Kinds of Set
1. Finite set is a set with countable number of elements. The number of elements in a
set is called cardinal number.
Example: The set of letters in the English Alphabet. Letting L be the name of this set
thus, n(L) = 26; the cardinal number.
2. Infinite set is a set that has uncountable number of elements.
Example: The set of counting numbers
3. Empty/Null/Void set is a set that has no elements, denoted by Ø or by a pair of
braces with no element inside, i.e. { }.
Example: The set of numbers in the Filipino alphabet
4. Universal set is a set that indicates the totality of all the elements under consideration,
denoted by U.
Example: The set of real numbers.
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Relationships Between Sets
1. Equal sets are sets with exactly the same elements.
Example:
Let A = { xx is the distinct letter of the word tame } and
B = { xx is the distinct letter of the word mate }
Therefore, sets A and B are equal sets, denoted by A = B, since both sets have the
elements a, e, m and t.
2. Equivalent sets are sets with the same cardinal number.
Example: Let C = { xx is a neither positive nor negative integer }
and let D = { xx is an even prime number }.
Since C = { 0 } and D = { 2 }, wherein both C and D have only one
element; then, C is equivalent to D, denoted by C  D.
Note: Equal sets are equivalent sets, however, not all equivalent sets are equal sets.
3. Joint sets are sets with at least one common element.
Example: Let E = {xx is the set of positive factors of 6}
and let F = {xx is the set of positive factors of 9}
Since E = {1, 2, 3, 6} and F = {1, 3, 9}, wherein elements 1 and 3 can be
found both in sets E and F, then these sets are joint sets.
4. Disjoint sets are sets that have no common element.
Example: Let G = {xx is the set of even numbers}
and let H = {x/x is the set of odd numbers}
Therefore, sets G and H are disjoint sets since no even number is an odd number
and vice versa.
5. Subset . For any sets A and B, A is said to be a subset of B, written in symbol A  B.
The symbol  means “a subset of” while  means “not a subset of”. If the first set
equals the second set, then it is an improper subset. The symbol  is used to mean
an improper subset.
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Note: Null and the set itself are always subsets of any given set and are considered
improper subsets of the given set. Other than the set itself and the null set, all are
considered proper subsets.
The set containing all the subsets of the given set with n number of elements is
called the power set with 2n number of elements.
Example: Let P = {xx is the last three letters in the English alphabet } = {x, y, z}
Since P has three elements, then the number of subsets can be computed, 23= 8
subsets.
Listing down the subsets of P, we have Ø, {x}, {y}, {z}, {x, y}, {x, z}, {y, z}, and
P. Sets {x}, {y}, {z}, {x, y}, {x, z}, {y, z} are proper subsets of P while the set P itself
and Ø are its improper subsets.
Wait! … pause for a while, answer first the following question.
SAQ1
If C = {c, l, s, u}, tell whether the given statement is true or false.
a.
c  {c, l, s, u}
b.
s  {c, l, s, u}
c.
  {c, l, s, u}
d.
0  
e.
{c, l, s, u} = {u, s, l, c}
f.
C  {c, l, s, u}
g.
c, s, u  C
h.
CC
i.
C  {C}
j.
C has 16 proper subsets
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ASAQ1
a. True
f. False
b. False
g. False
c. True
h. True
d. False
i. True
e. True
j. False
Operations on Sets
1. Union of Sets A and B [denoted by A  B] - set whose elements are found in A or B
or in both.
In symbol: A  B = { xx  A  x  B }
Example: Given A = { a, b, c, d } and
B = { c, d, e } then, A  B = { a, b, c, d, e }
2. Intersection of sets A and B [denoted by A  B] - set whose elements are
common to both sets.
In symbol: A  B = { xx  A  x  B }
Example: Given A = { a, b, c, d } and B = { c, d, e } then, A  B = { c, d }.
3. Difference of sets A and B [denoted by A – B] - the set whose elements are
found in set A but not in set B.
In symbol: A – B = { xx  A and x  B }
Example: Given A = { a, b, c, d } and B = { c, d, e } then, A – B = { a, b }
4. Complement of Set A [ denoted by A' ] - the set of elements found in the universal
set but not in set A.
In symbol:
A' = { xx  U and x  A }
Example: Given U = { a, b, c, d, e }, A = { a, b, c, d }, and B = { c, d, e } then,
A' = { e } and B' = { a, b }.
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Wait! … pause for a while, answer first the following question.
SAQ2
Suppose that
U = {0,1,2,3,4,5,6,7,8,9},
A = {0,2,4,6,8},
B = {1,3,5,7,9}, and
C = (3,4,5,6,7}.
Find:
a) (A  B)  C
c) (A – C’)  (B  C)
b) A  (C – B)
d) [ (A  B)’ – C’ ]’
ASAQ2
Solution:
a) (A  B)  C =   C = C = {3,4,5,6,7}
b) A  (C – B) = {0,2,4,6,8}  {4,6} = {4,6}
c) (A – C’)  (B  C) = {4,6}  {1,3,4,5,6,7,9} = {4,6}
d) [ (AB)’ – C’ ]’ = [’ – {0,1,2,8,9} ]’ = {3,4,5,6,7}’ = {0,1,2,8,9}
Venn-Euler Diagram
The pictorial representation of relationship and operations of sets is the so-called
Venn-Euler Diagrams, or simply Venn Diagrams. The universal set is usually
represented by a rectangle while circles within the rectangle usually represent its
subsets. The shaded region in the given diagrams illustrates the indicated sets relation
or sets operation.
AB
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AB
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A–B
A
A–B
A
Venn Diagrams are especially useful in solving problems such as in the following
examples.
A survey was conducted about the preferred mobile cell phone brand with 300 college
freshmen students as respondents and it was found out that
150 preferred Nokia (N)
125 preferred Motorola (M)
100 preferred Sony-Ericsson (S)
50 preferred Nokia and Motorola
40 preferred Nokia and Sony-Ericsson
35 preferred Motorola and Sony-Ericsson
20 preferred all three brands
Let the universal set U contain all the 300 college
freshmen students who were
surveyed and the three circles be the three mobile cell phone brands, i.e. Nokia,
Motorola and Sony-Ericsson. There were 20 who preferred the three brands; thus, 20
would be written in the region common to all the three circles.
Thirty-five preferred Motorola and Sony-Erickson; hence,
35 will be written in
the region common to Motorola and Sony Erickson. Since 20 can be found in this region,
then 35 – 20 = 15. Only 15 who preferred Motorola and Sony-Ericsson but not Nokia
would be written on the other part of the region common to Motorola and Sony-Ericsson.
The same thing would be done to the number of those who preferred Nokia and
Motorola but not Sony-Ericsson computed as 50 – 20 = 30 and Nokia and Sony-Erickson
but not Motorola, solved as 40 – 20 = 20.
For the circle associated with Sony-Ericsson, note that there were already 20 +
20 + 15 = 55. Then, 100 – 55 = 45 preferred Sony Ericsson only. For those who
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preferred Motorola only, 125 – (30 + 20 + 15) = 60 and for those who preferred
Nokia
only, 150 – (30 + 20 + 20) = 80.
Finally,
the remaining 30, those who do not prefer any
of the three brands,
would be written in the region outside these three circles.
Wait! … pause for a while, answer first the following question.
SAQ3
Analyze the survey problem below. Make a Venn diagram and answer the
preceding questions.
In a certain university, 100 students were asked regarding the type of novel
books they preferred to read. The survey revealed that
71 preferred Romance
62 preferred Mystery
53 preferred Science Fiction
44 preferred Mystery and Romance
35 preferred Romance and Science Fiction
26 preferred Science Fiction and Mystery
17 preferred all the three types of books
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How many students preferred:
1. exactly one type of book?
___________
2. at most two types?
___________
3. at least one type?
___________
4. all the three types?
___________
5. none of the three types?
___________
6. Science Fiction only?
___________
7. Romance and Mystery?
___________
8. Mystery or Romance?
___________
9. Mystery and Romance but not Science Fiction? ___________
10. Mystery or Science Fiction but not Romance? ___________
ASAQ3
Venn diagram
1. 27 exactly preferred 1 type of book.
2. 83 preferred at most 2 types of books.
3. 98 preferred at least 1 type of book
4. 17 preferred all the three types.
5. 2 did not prefer all the 3 types.
6. 9 preferred Science Fiction only.
7. 44 preferred Romance and Mystery.
8. 89 preferred Mystery or Romance.
9. 27 preferred Mystery and Romance but not Science Fiction.
10. 27 preferred Mystery or Science Fiction but not Romance.
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Classes of Sets, Power Sets
Given a set S, we may wish to talk about some of its subsets. Thus we would be
considering a “set of sets”. Whenever such situation arises, to avoid confusion, we will
speak of a class of sets. If we wish to consider some of the sets in a given class of sets,
then we will use the term subclass.
Example: Suppose S = {1, 2, 3, 4}. Let A be the class of subsets of S which contain
exactly three elements of S. Then
A = { {1,2,3}, {1,2,4}, {1,3,4}, {2,3,4} }
The elements of A are the sets {1,2,3}, {1,2,4}, {1,3,4}, and {2,3,4}. Let B be the class of
subsets of S which contain 2 elements. Then
B = { {1,2}, {1,3}, {1,4}, {2,3}, {2,4}, {3,4} }
The elements of B are {1,2}, {1,3}, {1,4}, {2,3}, {2,4}, {3,4}. Thus, neither A is a subclass
of B nor B is a subclass of A.
For a given set S , we may speak about the class of all subsets of S. This class is
called the power set of S, and it will be denoted by P(S). If S is finite, then so is P(S). In
fact, the number of elements in P(S) is 2 raised to the power of n(S); that is, n(P(S)) =
2n(S) .
Example: Suppose S = {1, 2, 3}. Then
P(S) = { , {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, S}
Note that the empty set  belongs to P(S) since  is a subset of S. Similarly S belongs
to P(S). As expected P(S) has 23 = 8 elements.
Axiomatic Development of Set Theory
Any axiomatic development of a branch of mathematics begins with the following:
1. undefined terms,
2. undefined relations; and
3. axioms relating the undefined terms and undefined relations.
Then, one develops theorems based upon the axioms and definitions.
Consider, for example, the axiomatic development of set theory and it begins
with the following:
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1. “element” and “set” are undefined terms;
2. “element belongs to a set” is the undefined relation.
Two of the axioms(called principles) of set theory follow:
Principle of Extension. Two sets A and B are equal if and only if they have the same
elements, that is if every element in A belongs to B and every element in B belongs to A.
Principle of Abstraction. Given any set U and any property P, there is a set A such that
the elements of A are exactly those elements in U which have the property P; that is,
A = { xx  U, P(x) is true}
There are other axioms which are not listed. As our treatment of set theory is
mainly intuitive, we will refrain from any further discussion of the axiomatic development
of set theory.
Wait! … pause for a while, answer first the following question.
SAQ4
Find the number of elements in the power set of each of the following sets:
a. {days of the week}
c. {seasons of the year}
b. {positive divisors of 12}
d. {letter in the word “yes”}
ASAQ4
a. 27 = 128.
b. 26 = 64 since there are six divisors, 1, 2, 3, 4, 6, 12, of 12.
c. 22 = 4 since there are two seasons.
d. 23 = 8.
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Activity No. 6
I. Let: U = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 },
A = { 1, 3, 4, 5, 6 }, B = { 4, 5, 9, 10 },C = { 1, 2, 3, 7, 8 }. Then,
1. A’ = _____________________________________________
2. A  B = ___________________________________________
3. B  C = __________________________________________
4. C – B = ___________________________________________
5. B – A = ___________________________________________
6. (C  B)’ = _________________________________________
7. (C  A)’ = _________________________________________
8. (A  B)  C = ______________________________________
9. (B  C)  A = ______________________________________
10.(C  A)’ – (C  B)’ = ________________________________
II. Let A be a subset of U, then
1. A  U = ___________________________________________
2. A  U = ___________________________________________
3. A   = __________________________________________
4. A   = __________________________________________
5.   U = __________________________________________
6. U   = __________________________________________
7. A  A’ = ___________________________________________
8. A  A’ = ___________________________________________
9. (A’)’
= ___________________________________________
10.’
= __________________________________________
11. U’
= ___________________________________________
12. U – A = ___________________________________________
13. U -  = ___________________________________________
14.  - A = ___________________________________________
15.  - U = ___________________________________________
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III. Given the figure below, shade the region representing each of the following
U
Uu
Cc
BB
BBBB
B
A
a. A  C
c. (A  B)  (B  C)
e. B – A
g. (A  B) 
b. A  C
d. .(A  B)  (B  C)
f. A – B
h. (A  C) 
IV. Determine the power set P(A) of A = {a, b, c, d}
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