Lesson 1 5 : Rela tions a nd algebr as

1. Lesson 15:Relations and algebras Mathematical Logic;

2. 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 Relations and algebras

3. 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 Relations and algebras

4. Naive theory of sets • Ø – an empty set • Cardinality of a set A: |A| Relations between sets (axioms): • Equality • Inclusion (being a subset) Relations and algebras

5. Naive theory of sets – set theoretical operations • Intersection • Union • Difference • Symetrical difference • Complement with respect to universe U Relations and algebras

6. Naive theory of sets – set theoretical operations • Power set • Cartesian product • Cartesian power A1=A, A0={Ø} n Relations and algebras

7. 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. Relations and algebras

8. 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): Relations and algebras

9. Binary relations • Inverse relation: • Composition of relations Relations and algebras

10. 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) ror (y,x) r Relations and algebras

11. Binary relations The important types of binary relations: • Tolerance – reflexive, symmetric • Quasi-ordering – reflexive, transitive • Equivalence – reflexive, symmetric, transitive • Partial ordering – reflexive, antisymmetric, transitive Relations and algebras

12. 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, ... Relations and algebras

13. Mappings (functions) • f A  B is called a mappingfrom 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 amapping 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 Relations and algebras

14. Mapping (functions) • Examples: 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 r A  B s A  B t A  B Relations and algebras

15. Mapping (functions) Mapping f: AB is called • Injection (one to one total mapping of A intoB), iff: x1,x2A, y B: (x1,y) f and(x2,y) f x1 = x2 • Surjection (mapping of A ontoB), iff: y B xA: (x,y) f • Bijection (one to onemapping of A ontoB)(mutually single-valued), iff it is injection and surjection. Relations and algebras

16. Mapping (functions) • Examples: j: ZZ, j(n)=n2, k: ZN, k(n)=|n|, l: NN, l(n)=n+1,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 Relations and algebras

17. Partitions and equivalences • Partition on a set A is a system such that:X = {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 Relations and algebras

18. 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 Ainduced by equivalence r; • where [x]ris 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}  X={X1, X2, X3} • X1={…-6, -3, 0, 3, 6, …} • X2={…-5, -2, 1, 4, 7, …} • X3={…-4, -1, 2, 5, 8, …} Relations and algebras

19. Orderings • If r is an order relation on A, then a couple (A,r) is called an ordered set • Examples: (N, ), (2M, ) • Coverrelation 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} Relations and algebras

20. Orderings • Hasse diagram –graphicalpicturing • Example: (A, ), A={a,b,c,d,e} r = {(a,b), (a,c), (a,d), (b,d)}idA idA={(a,a): aA} Relations and algebras

21. 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 Relations and algebras

22. 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 isotonef: 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 Relations and algebras

23. 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 Relations and algebras

24. 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 everyM  A, then (A, ) is called a complete lattice Relations and algebras

25. 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 Relations and algebras

26. 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,+) ... Relations and algebras

27. Algebras with one binary operation • We can define a finite grupoid G=(G,) byCayley table • Example: • G = {a,b,c} • For example: a  b = b, b  a = a, c  c = b ... Relations and algebras

28. 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) Relations and algebras

29. 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 Relations and algebras

30. 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 Gb 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 Hb = a Gb • Examples: • (N,+N) is a subgrupoid of (Z,+Z) Relations and algebras

31. Algebras with one binary operation • Let G =(G, G), H =(H, H) begrupoids and h:GHbe a mapping. Thenhis a homomorphismofthegrupoidGintothegrupoidH, ifa,bG: h(a Gb) = h(a) Hh(b) Typesofhomomorphism: • Monomorphism – hisinjective • Epimorphism – hissurjective • Isomorphism – hisbijective • Endomorphism – H=G • Automorphism – isbijective and H=G Examples: • (R,+), h(x)= -x , his automorfismus (R,+) intoitself • h(x+y) = -(x+y) = (-x) + (-y) = h(x) + h(y) Relations and algebras

32. 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 Gb1, a2 Gb2)  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 Gb]r Example: • r ZZ; r = {(x,y); 3 divides x-y} • r is a congruence on (Z,+) Relations and algebras

33. Algebras with one binary operation Typesofgrupoids: • Semigroup– anassociativegrupoid • Monoid– a semigroupwiththeneutral element • Group – a monoidwiththe inverse elements (i.e., theoperationisassociative, thereistheneutral element, and to each element thereisaninverseone) • Abeliangroup– a commutativegroup Examples: (Z isthe set ofintegers, N isthe set ofnaturals) • (Z, –) – grupoid, not a semigroup • (N – {0}, +) – semigroup, not a monoid • (N, ) – monoid, not a group • (Z, +) – Abeliangroup, • neutral element is 0, inverse element is –x (minus x) • (Z, ) – Abeliangroup, • neutral element is 1, inverse element is 1/x Relations and algebras

34. 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·(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 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 ringunit (A,+,·). Relations and algebras

35. Algebras with two binary operation A ring (A,+,·) is called afield, if: • (A - {0},·) is a commutative group Examples: • (Z,+,·) – a ring, not a field • (R,+,·),  (C,+,·) – fields Relations and algebras

36. Lattice – algebraic structure • Lattice L = (L, , ) • : L  LL, : L  LL • x, y, z  L it holds: Relations and algebras

37. 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 = sup{a,b}, a b = inf{a,b}, Then the followingholds: • (A, ) is a lattice ordered set, where: sup{a,b}= a  b, inf{a,b}= a  b • (B, , ) is a lattice • (A, , ) = (A, , ) Relations and algebras

38. Lattice – algebraic structure A lattice (L, , ) is called: • Modular, ifitholds: x, y, z  L : x  z x  (y  z) = (x  y)  z • Distributive, ifitholds: x, y, z  L : x  (y  z) = (x  y)  (x  z) x  (y  z) = (x  y)  (x  z) • Complementary, ifitholds: : Thereisthe least element 0  L and thegreatest element 1  L, and x  L x’  L : x  x’ = 0, x  x’= 1 x’ iscalled a complementofan element x Relations and algebras

39. 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 Relations and algebras

40. Lattice – algebraic structure Lattice (L, , ) is called a Boolean lattice, if it is: • Complementary, distributive, with the least element 0  L and with thegreatest 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} Relations and algebras