Lesson 15: Relations and algebras

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Lesson 15Relations and algebras

• Mathematical Logic

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Contents

• Theory of sets
• Relations and mappings
• 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), ?
  (union)
• 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
• Inclusion (being a subset)

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Naive theory of sets set theoretical operations

• Intersection
• Union
• Difference
• Symetrical difference
• Complement with respect to universe U

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Naive theory of sets set theoretical operations

• Power set
• Cartesian product
• Cartesian power
• A1A, A0Ø

n
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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
• 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)

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Binary relations

• Inverse relation
• Composition of relations

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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
  ? xy
• 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 xy 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
• u (x,y)?Z?Z xy2, v (x,y)?N?N xy2,
• w (x,y)?Z?Z yx2
• r, u are not mappings
• s, v are partial mappings from A to B, (not
  total) t, w are total mappings

r ? A ? B
s ? A ? B
t ? A ? B
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Mapping (functions)

• Mapping f A?B is called
• Injection (one to one total mapping of A into B),
  iff ?x1,x2 ? A, ?y ? B
• (x1,y) ? f and (x2,y) ? f ? x1 x2
• Surjection (mapping of A onto B), iff
• ?y ? B ?x ? A (x,y) ? f
• Bijection (one to one mapping of A onto B)
  (mutually single-valued), iff it is
• injection and surjection.

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Mapping (functions)

• Examples
• j Z?Z, j(n)n2, k Z?N, k(n)n,
• l N?N, l(n)n1, 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

f A ? B
g A ? B
h A ? B
i A ? B
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Partitions and equivalences

• Partition on a set A is a system such thatX
  Xi i ? I
• Xi ? A for ?i ? I
• Xi ?Xj Ø for ?i,j ? I, i ? j
• U X A (the union of Xi A)
• Xi classes of the partition
• Fine graining of a partition X Xi i ? I
  is a system such that
• 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 xr x ? A a partition on A induced by
  equivalence r
• where xr is the class of those elements that
  are equivalent with x
• quotient set (factor set) of the set A induced by
  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 ?
  XX1, 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
• Examples (N, ?), (2M, ?)
• Cover relation
• Let (A, ?) be an ordered set, (a,b)?A
• a lt b (b covers a), iff
• a lt b and ??c ?A a ? c a c ? b
• Examples (N, ?), lt (n,n1) n ? N

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Orderings

• Hasse diagram graphical picturing
• Example
• (A, ?), Aa,b,c,d,e
• r (a,b), (a,c), (a,d), (b,d) ? idA
• idA(a,a) a?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)
• Example
• The least does not exist
• The greatest does not exist
• Minimal a, e
• Maximal d, c, e

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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 of M in A
• UA(M)x?A ?m?M m ? x
• The set of upper bounds 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-like
  ordered set), iff
• ?x,y?A ?s,i ?A s sup(x,y), i inf(x,y)
• Notation
• x ? y sup(x,y) (join)
• x ? y inf(x,y) (meet)
• 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 the algebra
• FA fi Ap(fi)?A i?I a set of operations
  (functions, mappings) 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 define a finite grupoid G(G,?) by Cayley
  table
• Example
• G a,b,c
• For example a ? b b, b ? a a, c ? c b
  ...

? a b c
a a b c
b a c c
c a a 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 (ab) / 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) (13, 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 hG?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 HG
• Automorphism is bijective and HG
• Examples
• (R,), h(x) -x , h is automorfismus (R,) into
  itself
• h(xy) -(xy) (-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), ar ?G/r br a ?G br
• Example
• r ? Z?Z r (x,y) 3 divides x-y
• r is a congruence on (Z,)

? 0 1 2
0 0 1 2
1 1 2 0
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 (i.e.,
  the operation is associative, there is the
  neutral element, and to each element there is an
  inverse one)
• Abelian group a commutative group
• Examples (Z is the set of integers, N is the set
  of naturals)
• (Z, ) grupoid, not a semigroup
• (N 0, ) semigroup, not a monoid
• (N, ?) monoid, not a group
• (Z, ) Abelian group,
• neutral element is 0, inverse element is x
  (minus x)
• (Z, ?) Abelian group,
• neutral element is 1, inverse element is 1/x

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Algebras with two binary operation

• Algebra (A,,) is called a Ring, if
• (A,) is a commutative group
• (A,) is a monoid
• For ?a,b,c?A holds a(bc)abac,
  (bc)aba ca
• If Agt1, then (A,,) is called a non-trivial
  ring.
• Let 0 ?A is the neutral element of the group
  (A,). Then 0 is called the ring zero (A,,).
• Let 1?A is the neutral element of the 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, 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 associativity
x ? (x ? y) x x ? (x ? y) x 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 operations meet and join ?? and
  ?? on B
• a ?? b supa,b, a ?? b infa,b,
• Then the following holds
• (A, ?) is a lattice ordered set, where
• supa,b a ? b, infa,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, and
• ?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 the smallest non modular
  lattice

M5
N5
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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