Lesson 14:
Relations and algebras
Mathematical Logic;
advanced course
Contents
 Theory of sets (revision)
 Relations and mappings (revision)
–
–
–
–
–
Relation
Binary relation on a set
Mapping
Partition & equivalence
Orderings
 Algebras
– Algebras with one operation
– Algebras with two operations
– Lattices
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Theory of sets
 Language:
• Special symbols:
– Binary predicates:  (is an element of),  (is a proper
subset of),  (is a subset of)
– Binary function symbols:  (intersection – „průnik“),
 (union – „sjednocení“)
 Cantor – naive set theory (without axiomatizing)
 Nowadays there are many formal
axiomatizations, but none of them is complete.
 Examples: von Neumann-Bernays-Gödel,
Zermelo-Fränkel + axiom of choice
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Naive theory of sets
 Ø – an empty set
 Cardinality of a set A: |A|
Relations between sets (axioms):
 Equality
A  B, iff (x : x  A  x  B)
 Inclusion
A  B, iff (x : x  A  x  B)
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Naive theory of sets – set theoretical operations
 Intersection
U
 Union
U
 Difference
U
A  B  {x; x  A and x  B}
A  B  {x; x  A or x  B}
A  B  {x; x  A and x  B}
 Symetrical difference
A  B  ( A  B)  ( A  B)
A
B
A
B
A
B
A
B
U
 Complement with respect to universe U U
A'  U  A
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Naive theory of sets – set theoretical operations
 Power set
2 A  {X ; X  A}
 Cartesian product
A  B  {(a, b) : a  A, b  B}
 Cartesian power
An  A  A   A
n
A1=A, A0={Ø}
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Zermelo-Fränkel set-theory
Axiom of extensionality: Two sets are identical if and only if they have the same
elements.
Axiom of an empty set: There is a set with no elements.
Axiom of pairing: If x, y are sets, then so is {x,y}, a set containing x and y as its
only elements.
Axiom of union: Every set has a union. That is, for any set x there is a set y
whose elements are precisely the elements of the elements of x.
Axiom of infinity: There exists a set x such that {} is in x and whenever y is in x,
so is the union y U {y}.
Axiom of separation (or subset axiom): Given any set and any proposition P(x),
there is a subset of the original set containing precisely those elements x
for which P(x) holds.
Axiom of replacement: Given any set and any mapping, formally defined as a
proposition P(x,y) where P(x,y) and P(x,z) implies y = z, there is a set
containing precisely the images of the original set's elements.
Axiom of power set: Every set has a power set. That is, for any set x there exists
a set y, such that the elements of y are precisely the subsets of x.
Axiom of regularity (or axiom of foundation): Every non-empty set x contains
some element y such that x and y are disjoint sets.
Axiom of choice: (Zermelo's version) Given a set x of mutually disjoint nonempty
sets, there is a set y (a choice set for x) containing exactly one element
from each member of x.
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Relations
 n-ary relation between the sets A1, A2, ..., An
r  A1  A2   An
 Examples:
– D = a set of possible days
– M = a set of VŠB rooms
– Z = a set of VŠB employees
A ternary relation meeting (when, where, who):
r  D M 2
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Binary relations
r  A B
 Inverse relation:
r
1
 {(x, y)  B  A : ( y, x)  r}
 Composition of relations
r  A  B, s  B  C
r  s  {( x, y)  A  C : (z  B) ( x, z)  r and ( z, y)  s  }
A
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Binary relations
Binary relation r on a set A is called:
 Reflexive: x  A: (x,x)  r
 Irreflexive: x  A: (x,x)  r
 Symmetric: x,y  A: (x,y)  r  (y,x)  r
 Antisymmetric: x,y  A: (x,y)  r and (y,x)  r  x=y
 Asymmetric: x,y  A: (x,y)  r  (y,x)  r
 Transitive: x,y,z  A: (x,y)  r and (y,z)  r  (x,z)  r
 Cyclic: x,y,z  A: (x,y)  r and (y,z)  r  (z,x)  r
 Continuous: x,y  A: x=y or (x,y)  r or (y,x)  r
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Binary relations
The important types of binary relations:
 Tolerance – reflexive, symmetric
 Quasi-ordering – reflexive, transitive
 Equivalence – reflexive, symmetric,
transitive
 Partial ordering – reflexive, antisymmetric,
transitive
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Binary relations
Examples:
 Tolerance:
– „being similar to“ on a set of people,
– „being no more than one year older or younger“ on the set of
people, ...
 Quasi-ordering:
– on a set of sets: the sets X and Y are related iff |X||Y|,
– divisibility relation on the set of integers,
– „not to be older“ on a set of people, ...
 Equivalence:
– „being of the same age“ on the set of people,
– identity on the set of natural numbers, ...
 Ordering:
– Set-theoretical inclusion  on a set of sets,
– divisibility relation on the set of natural numbers, ...
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Mappings (functions)
 f  A  B is called a mapping from a set A
into a set B (partial mapping), iff:
(x  A, y1,y2  B)
( (x,y1)  f and (x,y2)  f  y1 = y2 )
 f is called a mapping of a set A into a set
B (total mapping, written as: f: AB), iff:
– f is mapping from A to B
– (x  A)(y  B) ( (x,y)  f)
 If f is a mapping and (x,y)  f, then we write
f(x)=y
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Mapping (functions)
 Examples:
rAB
A
B
sAB
A
B
tAB
A
B
u = {(x,y)ZZ; x=y2}, v = {(x,y)NN; x=y2},
w = {(x,y)ZZ; y=x2}
 r, u – are not mappings
 s, v – are partial mappings from A to B, (not total)
t, w – are total mappings
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Mapping (functions)
Mapping f: AB is called
 Injection (one to one total mapping A into B), iff:
x1,x2  A, y  B:
(x1,y)  f and (x2,y)  f  x1 = x2
 Surjection (mapping A onto B), iff:
y  B x  A: (x,y)  f
 Bijection (one to one mapping A onto B)
(mutually single-valued), iff it is
injection and surjection.
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Mapping (functions)
 Examples:
f:AB
A
B
g:AB
h:AB
i:AB
A
A
A
j: ZZ, j(n)=n2,
l: NN, l(n)=n+1,




B
B
B
k: ZN, k(n)=|n|,
m: RR, j(x)=x3
f, j – are neither injections nor surjections
h, k – are surjections, not injections
g, l – are injections , not surjections
I,m – are injections and surjections simultaneously 
bijections
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Partitions and equivalences
 Partition on a set A is a system such that:
X = { Xi; i  I }
A
– Xi  A for i  I
– Xi Xj = Ø for i,j  I, i  j
– UX=A
Xi – classes of the partition
 Fine graining of partition X = { Xi; i  I } is a
system such that:
A
Y = { Yj; j  J }, iff:
– j  J, i  I so that Yj  Xi
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Partitions and equivalences
 Let r be an equivalence relation on a set A,
X is a partition on A, then it holds:
 Xr = {[x]r; x  A} – a partition on A (rozklad) induced by
equivalence r;
– a quotient set (faktorová množina) of the set A according to the
equivalence r
 rX = {(x,y); x and y belong to the same class of the
partition X}
– equivalence on A (induced by the partition X)
Example:
 r  ZZ; r = {(x,y); 3 divides x-y }  X={X1, X2, X3}
– X1={…-6, -3, 0, 3, 6, …}
– X2={…-5, -2, 1, 4, 7, …}
– X3={…-4, -1, 2, 5, 8, …}
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Orderings
 If r is an order relation on A, then a couple
(A,r) is called an ordered set (uspořádaná
množina)
– Examples: (N, ), (2M, )
 Cover relation (relace pokrytí)
Let (A, ) be an ordered set, (a,b)A
a –< b („b covers a“), iff:
a < b and c A: a  c a c  b
– Examples: (N, ), –< ={(n,n+1); n  N}
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Orderings
 Hasse diagram – graphical picturing
– Example:
(A, ), A={a,b,c,d,e}
r = {(a,b), (a,c), (a,d), (b,d)}  idA
idA={(a,a): aA}
d
b
c
e
a
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Orderings
An element a of an ordered set (A, ) is called:
 The least: for xA: a  x
 The greatest: for xA: x  a
 Minimal: for xA: (x  a  x = a)
 Maximal: for xA: (a  x  x = a)
d
 Example:
–
–
–
–
The least: does not exist
The greatest: does not exist
Minimal: a, e
Maximal: d, c, e
b
c
e
a
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Orderings
 A mapping of the ordered sets (A, ), (B, ) is
called isomorphic, iff there is a bijection f: AB
such that:
x,yA: x  y, iff f(x)  f(y)
 A mapping of the ordered sets (A, ), (B, ) is
called isotone f: AB, when the following holds:
x,yA: x  y  f(x)  f(y)
 Examples:
 f: NZ, f(x)=kx, k Z, k  0 is isotone
 g: NZ, g(x)=kx, k Z, k  0 is not isotone
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Orderings
 Let (A, ) be an ordered set, M  A, then
 LA(M)={xA; mM: x  m }
– The set of lower bounds (dolní závory) of M in A
 UA(M)={xA; mM: m  x }
– The set of upper bounds (horní závory) of M in A
 InfA(M) – The greatest element of the set LA(M)
– Infimum of the set M in A
 SupA(M) – The least element of the set UA(M)
– Supremum of the set M in A
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Lattice
defined as an ordered set
 A set (A, ) is called a lattice (lattice ordered
set), iff:
x,yA s,i A : s = sup({x,y}), i = inf({x,y})
 Notation:
– x  y = sup({x,y})
– x  y = inf({x,y})
(spojení – „meet“)
(průsek – „join“)
 If sup(M) and inf(M) exist for every M  A,
then (A, ) is called a complete lattice
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Algebra
Algebra (abstract algebra) is a couple: (A, FA):
 A  Ø – an underlying set of algebra (nosič)
 FA = {fi: Ap(fi)A; iI} – a set of operations on A
 p(fi) – the arity of an operation fi
 Examples:
– (N, +2, 2)
the set of natural numbers with the addition and
multiplication operations
– (2M, , )
the set of all subsets of a set M with the intersection
and union operations
– (F, , )
The set (F) of the propositional logic formulas with the
conjunction and disjunction operations
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Algebras with one binary operation
Grupoid G=(G,)
 : G  G  G
 If a set G is finite, then the grupoid G is
called finite
 Order of a grupoid = |G|
Examples of grupoids:
 G1=(R,+), G2=(R,), G3=(N,+) ...
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Algebras with one binary operation
 We can express the finite grupoid G=(G,)
by Cayley table
 Example:
 G = {a,b,c}

a
b
c
a
a
b
c
b
a
c
c
c
a
a
b
 For example: a  b = b, b  a = a, c  c = b
...
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Algebras with one binary operation
Let G=(G,) be a grupoid, then G is:
 Commutative, if
– (a,bG)(a  b = b  a)
 Associative, if
– (a,b,cG)((a  b)  c = a  (b  c))
 With a neutral element, if
– (e G aG)(a  e = a = e  a)
 With an aggressive element, if
– (o G aG)(a  o = o = o  a)
 With an inverse elements, if
– (aG b G)(a  b = e = b  a)
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Algebras with one binary operation
Examples:
 (R,), (N,+) – commutative and associative
 (R,), a  b = (a+b) / 2 – commutative, not associative
 (R,), a  b = ab
– neither commutative nor associative
 (R,) – 1 = the neutral element, 0 = the aggressive
element
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Algebras with one binary operation
Let G=(G,G) be a grupoid. HG is called closed
with respect to the operation G, if
 (a,bH)(a G b H)
A Grupoid H=(H,H) is a subgrupoid of a grupoid
G=(G,G), if it holds:
 Ø  H  G is closed
 a,bH: a H b = a G b
 Examples:
 (N,+N) is a subgrupoid of (Z,+Z)
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Algebras with one binary operation
 Let G1=(G1, 1), G2=(G2, 2) be grupoids.
 G1  G2 =(G1  G2, ) – direct product G1 and G2,
where:
 (a1, a2)  (b1, b2) = (a1 1 b1, a2 2 b2)
Examples:
 G1=(Z,+), G2=(Z,).
 G1  G2 =(Z  Z, ),
 (a1, a2)  (b1, b2) = (a1 + b1, a2  b2)
 (1,2)(3,4) = (1+3, 2  4) = (4,8) and so on.
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Algebras with one binary operation
 Let G =(G, G), H =(H, H) be grupoids and h:GH be a mapping.
Then h is a homomorphism of the grupoid G into the grupoid H, if
a,bG: h(a G b) = h(a) H h(b)
Types of homomorphism:
 Monomorphism – h is injective
 Epimorphism – h is surjective
 Isomorphism – h is bijective
 Endomorphism – H=G
 Automorphism – is bijective and H=G
Examples:
 (R,+), h(x)= -x , h is automorfismus (R,+) into itself
 h(x+y) = -(x+y) = (-x) + (-y) = h(x) + h(y)
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Algebras with one binary operation
r is called a congruence on a grupoid G=(G,G), iff:
 r is a binary relation on G: r  GG
 r is an equivalence
 (a1, a2), (b1, b2)  r  (a1 G b1, a2 G b2)  r
The factor grupoid of a grupoid G induced by the
congruence r:
G/r=(G/r,G/r), [a]r G/r [b]r = [a G b]r
 [0] [1] [2]
[0] [0] [1] [2]
Example:
 r  ZZ; r = {(x,y); 3 divides x-y }
[1] [1] [2] [0]
 r is a congruence on (Z,+)
[2] [2] [0] [1]
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Algebras with one binary operation
Types of grupoids:
 Semigroup – an associative grupoid
 Monoid – a semigroup with the neutral element
 Group – a monoid with the inverse elements
 Abelian group – a commutative group
Examples:
 (Z, –) – grupoid, not a semigroup
 (N – {0}, +) – semigroup, not a monoid
 (N,) – monoid, not a group
 (Z, +) – Abelian group
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Algebras with two binary operation
Algebra (A,+,·) is called a Ring (okruh), if:
 (A,+) is commutative group
 (A,·) is monoid
 For a,b,cA it holds:
a·(b+c)=a·b+a·c, (b+c)·a=b·a + c·a
 If |A|>1, then (A,+,·) is called a non-trivial ring.
 Let 0 A is the neutral element of group (A,+).
Then 0 is called the ring zero (A,+,·).
 Let 1A is the neutral element of monoid (A,·).
Then 1 is called the ring unit (A,+,·).
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Algebras with two binary operation
A ring (A,+,·) is called a field (pole), if:
 (A - {0},·) is a commutative group
Examples:
 (Z,+,·) – a ring, not a field
 (R,+,·), (C,+,·) – fields
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Lattice – algebraic structure
 Lattice L = (L, , )
 : L  LL,
: L  LL
 x, y, z  L it holds:
xx=x
xx=x
idempotention
xy=yx
xy=yx
commutativity
x (y  z) = (x  y)  z
x (y  z) = (x  y)  z
x (x  y) = x
x (x  y) = x
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associativity
absorption
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Lattice – algebraic structure
Let (A, , ) be a lattice, (B, ) be a lattice ordered set
 Let us define a relation  on A :
a  b, iff a  b = b
 Let us define the relations  and  on B
a  b = sup{a,b}, a  b = inf{a,b},
Then it holds:
 (A, ) is a lattice ordered set, where:
sup{a,b}= a  b, inf{a,b}= a  b
 (B, , ) is a lattice
 (A, , ) = (A, , )
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Lattice – algebraic structure
A lattice (L, , ) is called:
 Modular, if it holds: x, y, z  L :
x  z  x  (y  z) = (x  y)  z
 Distributive, if it holds: x, y, z  L :
x  (y  z) = (x  y)  (x  z)
x  (y  z) = (x  y)  (x  z)
 Complementary, if it holds: :
There is the least element 0  L and the greatest element
1L
x  L x’  L : x  x’ = 0, x  x’ = 1
x’ is called a complement of an element x
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Lattice – algebraic structure
 Each distributive lattice is modular
Examples:
 M5 (diamond) – a modular lattice which is not
distributive
 N5 (pentagon) – is not modular
1
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Lattice – algebraic structure
Lattice (L, , ) is called a Boolean lattice, if it is:
 Complementary, distributive, with the least element 0  L
and with the greatest element 1  L
Boolean algebra:
 (L, , , –, 0, 1), – : LL is an operation of complement
in L
Example:
 (2A, , ), A = {1,2,3,4,5,6,7}
9.4.2015
Relace a algebry
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