1. 2007 D iscrete The foundations C ounting theory N umber theory G raphs & trees structure s 2110200

2. Evaluation The foundations15 % Counting theory15 % Number theory15 % Graphs & trees15 % Final40 %

3. Motivation 34678532 27851484+ --- 62530016 How fast can we do the computation ?

4. Motivation ? A IJ B C G ED H F

5. K = 0 FOR I 1 = 0 TO M { FOR I 2 = 0 TO I 1 { FOR I 3 = 0 TO I 2 { … FOR I n = 0 TO I n-1 { K = K + 1 } … } Find the value of K

6. s T he Foundation Relational structures T s

7. The Foundations Logic and reasoning Set, relation and function Methods of proof

8. Logic & Reasoning

9. Historical SYLLOGISTIC REASONING Aristotle (384-322 B.C.) Euclid of Alexandria (325-265 B.C.) DEDUCTIVE REASONING Chrysippus of Soli (279-206 B.C.) MODAL LOGIC George Boole (1815-1864 A.D.) PROPOSITIONAL LOGIC Augustus De Morgan (1806-1871 A.D.) DE MORGAN’s LAWs

10. Propositional logic Definition A proposition is a declarative statement that is either true or false but not both.

11. S ummary T heorem : Logical Equivalences, given any propositions p,q and r, a tautology T and a contradiction C, the following logical equivalences hold: Commutative laws: p  q  q  p p  q  q  p Associative laws:( p  q )  r  p  ( q  r ) ( p  q )  r  p  ( q  r ) Distributive laws: p  ( q  r )  ( p  q )  ( p  r ) p  ( q  r )  ( p  q )  ( p  r ) Identity laws: p  T  pp  C  p Domination laws:( Universal bound laws ) p  T  Tp  C  C Idempotent laws: p  p  pp  p  p Negation laws: p  p  Tp  p  C Double negative laws:  (  p )  p De Morgan’s laws:  ( p  q )   p  q  ( p  q )   p  q Absorption laws: p  ( p  q )  p p  ( p  q )  p

12. Example Is the following assertion a proposition? This statement is false. No, since this statement is neither “true” nor “false”.

13. Example The n th statement in a list of 100 statements is What conclusion can you draw from these statements? Exactly n statements are false. At most one statement can be true, then 99 statements are false. That is only the 99 th statement is true.

14. Exactly n statements are false. Example The n th statement in a list of 100 statements is What conclusion can you draw from these statements? At least n statements are false. 50 first statements are true. The others are false.

15. Exactly n statements are false. Example The n th statement in a list of 100 statements is What conclusion can you draw from these statements? At least n statements are false. 99 CONTRADICTION

16. Conditional statement Converse The converse of p  q is q  p. Inverse The inverse of p  q is  p   q. Only if p only if q means If  q then  p. p  qp  q

17. Valid arguments An argument is valid means that if all hypotheses are true, the conclusion is also true. Definition An argument is a sequence of statements. All statements excluded the final one are called “hypotheses”, the final statement is called “conclusion”. A argument is the form: p ; q ; r ;…  f (read therefore)

18. Example Given an argument p  (q  r) ;  r ;  p  q pqrqrqrp  (q  r) rrp  q TTTTTFT TTFTTTT TFTTTFT TFFFTTT FTTTTFT FTFTTTT FFTTTFF FFFFFTF TRUE VALID Valid arguments ( p 1  p 2  p 3  …  p n )  q

19. Rules of inference Disjunctive addition Conjunctive simplification Conjunction addition Modus ponens Modus tollens Hypothetical syllogism Disjunctive syllogism Resolution Dilemma P  P  Q

20. Rules of inference P  Q  P Disjunctive addition Conjunctive simplification Conjunction addition Modus ponens Modus tollens Hypothetical syllogism Disjunctive syllogism Resolution Dilemma

21. Rules of inference P Q  P  Q Disjunctive addition Conjunctive simplification Conjunction addition Modus ponens Modus tollens Hypothetical syllogism Disjunctive syllogism Resolution Dilemma

22. Rules of inference P  Q P  Q Disjunctive addition Conjunctive simplification Conjunction addition Modus ponens Modus tollens Hypothetical syllogism Disjunctive syllogism Resolution Dilemma

23. Rules of inference P  Q  Q   P Disjunctive addition Conjunctive simplification Conjunction addition Modus ponens Modus tollens Hypothetical syllogism Disjunctive syllogism Resolution Dilemma

24. Rules of inference P  Q Q  R  P  R Disjunctive addition Conjunctive simplification Conjunction addition Modus ponens Modus tollens Hypothetical syllogism Disjunctive syllogism Resolution Dilemma

25. Rules of inference P  Q  P  Q Disjunctive addition Conjunctive simplification Conjunction addition Modus ponens Modus tollens Hypothetical syllogism Disjunctive syllogism Resolution Dilemma

26. Rules of inference P  Q  P  R  Q  R Disjunctive addition Conjunctive simplification Conjunction addition Modus ponens Modus tollens Hypothetical syllogism Disjunctive syllogism Resolution Dilemma

27. Rules of inference P  Q P  R Q  R  R Disjunctive addition Conjunctive simplification Conjunction addition Modus ponens Modus tollens Hypothetical syllogism Disjunctive syllogism Resolution Dilemma

28. Example Given two logical operators, p | qmeans  ( p  q ) p  qmeans  ( p  q ) Find a simple proposition for ( p  q )  ( p  q ). ( p  q )

29. Example Given two logical operators, p | qmeans  ( p  q ) p  qmeans  ( p  q ) Find a simple proposition for ( p  q )  ( p  q ). Find a proposition equivalent to p  q using only . (( p  p )  q )  (( p  p )  q )

30. Problem C onsider the following statements: All students go to school. John is a student. Diana is a student. …… O f course we can conclude that John goes to school. Diana goes to school. ……

31. Predicate logic The statement “All students go to school” has two parts: Variable students (denoted by variable x) “go to school” (the predicate) This statement can be denoted by P(x), where P denotes the predicate “go to school”. P(x) is said to be the value of the propositional function P at x. Once a value has been assigned to the variable x, the statement P(x) becomes a proposition and has a truth value.

32. Quantifiers Universal quantification  Existential quantification  Unique existential quantification  ! Consider a statement  x P(x)  Q(x). Contraposition Its contrapositive is  x  Q(x)   P(x). Inverse Its inverse is  x  P(x)   Q(x).. Converse Its converse is  x Q(x)  P(x).

33. Consider a statement  x P(x)  Q(x). For a particular e, P(e) is true, therefore Q(e) is true. Universal Modus Ponens

34. Consider a statement  x P(x)  Q(x). For a particular e,  Q(e) is true, therefore  P(e) is true. Universal Modus Tollens

35. Universal instantiation  xP(x)  P(c) if c  U. Universal generalization P(c) for an arbitrary c  U   xP(x) Existential instantiation  xP(x)  P(c) for some element c  U Existential generalization P(c) for some element c  U   xP(x) Rules of inferences

36. The order of quantifiers Given a predicate P(x,y): x + y = 0  x  y P(x,y)  y  x P(x,y)

37. Nested quantifiers  x P(x)   y Q(y)  x  y ( P(x)  Q(y) )  y  x ( P(x)  Q(y) )  x P(x)   x Q(x)  x  y ( P(x)  Q(y) ) Prenex normal form (PNF)

38. Express the following theorem using the first order predicate logic. Example Mathematical induction

39. Set George Cantor (1845-1918)

40. Set Definition A set is an unordered collection of objects. The objects are called the elements or members of the set. The number of distinct elements in a set is the cardinality of the set. The Cartesian product of A and B, denoted by A  B, is described by { (a,b) | a  A  b  B }.

41. Defining sets L 1 ={ n| for n = 1 2 3 … } L 2 ={ n | for n = 1 3 5 7 … } L 3 ={ n | for n = 1 4 9 16 … } L 4 ={ n | for n = 3 4 8 22 … }. Membership problem

42. Machine model M input output { yes, no } { space of input } { space of yes-input } MEMBERSHIP DECISION

43. Example Let U be the universe described by U = { x | 1000  x  9999}. Let A i be the set of all numbers in U such that the i th position is i. Find the cardinality of the union of A 1 A 2 A 3 and A 4 ?

44. Example Let S be the set of all x that x does not contain x. S = { x | x  x } Note that x is also a set. Does S contain S ? Russell’s paradox Bertrand Russell (1872-1970)

45. Set Operators UnionMutually disjoint IntersectionPartition Different Disjoint Complement Power set

46. Set Theorem Given sets A,B and C. Commutative laws: A  B = B  AA  B = B  A Associative laws: (A  B)  C = A  (B  C) Distributive laws: A  (B  C) = (A  B)  (A  C) Idempotent laws: A  U = AA  U = U De Morgan’s laws: (A  B) c = A c  B c (A  B) c = A c  B c Alternative representation for set difference A-B = A  B c Absorption laws: A  (A  B) = A(A  B)  A = A

47. Example The symmetric difference of A and B, ( A  B ), is the set containing those elements in either A or B, but not in both A and B. ( A  ( B  C ) )= ( ( A  B )  C ) ? YES

48. Example The symmetric difference of A and B, ( A  B ), is the set containing those elements in either A or B, but not in both A and B. ( A  ( B  C ) )= ( ( A  B )  C ) ? YES Given A  C = B  C Must it be the case that A = B ?

49. Multisets Definition Multisets are unordered collections of elements where an element can occur as a member more than once. { m 1.a 1, m 2.a 2, m 3.a 3, …, m r.a r } m i are called the multiplicities of the elements a i. OPERATORS: UNION, INTERSECTION, DIFFERENCE, SUM

50. Multisets Definition Multisets are unordered collections of elements where an element can occur as a member more than once. { m 1.a 1, m 2.a 2, m 3.a 3, …, m r.a r } m i are called the multiplicities of the elements a i. OPERATORS: UNION, INTERSECTION, DIFFERENCE, SUM Fuzzy sets 0  m i  1 Degree of membership

51. Relations & functions

52. Binary relation Definition Let A,B be sets. A binary relation R from A to B is a subset of the Cartesian product A  B. Given (x,y), ordered pair, in A  B, x is related to y by R, written xRy, iff (x,y)  R. Example (the congruence modulo 2 relation) The relation R from Z to Z is defined as follows; for all (x,y)  Z  Z, xRy iff x-y is even. Example, 6R2, 120R36 etc.

53. Function Definition A function F from A to B is a relation from A to B, F : A  B, that satisfies the following properties: For every x  A, there exists y  B such that (x,y)  F. For all x  A, and y, z  B, if (x,y)  F and (x,z)  F then y=z. For (x,y)  F, we usually write y =F (x) = image of x under F, and x is called pre-image of y under F. A is called domain of F. B is called co-domain of F. The set of all images of F is called range of F.

54. Compositions of functions Definition A function f, g from A to B is a function from A to B. (f + g)(x) = f(x) + g(x). (f g)(x) = f(x) g(x) The composition of the functions f and g, denoted by f  g, is defined as (f  g)(x) = f (g ( x ) )

55. Function Arrow diagram A function F from A to B. 12341234 abcdeabcde F(1) = b F(2) = a F(3) = b F(4) = e A B

56. Injective function Definition A function F from A to B is injective (or one-to-one) iff for all elements x and y in A, if F (x] = F (y] then x = y. Or, equivalently, if x  y then F (x]  F (y]. 12341234 abcdeabcde A B This function is not One-to-one.

57. Injective function Definition A function F from A to B is injective (or one-to-one) iff for all elements x and y in A, if F (x] = F (y] then x = y. Or, equivalently, if x  y then F (x]  F (y]. This function is an One-to-one. 12341234 abcdeabcde A B

58. Definition A function F from A to B is surjective (or onto) iff for any element y in B, it is possible to find an element x in A such that y = F (x]. 12341234 ab eab e A B This function is Onto. Surjective function

59. Definition A one-to-one correspondence (or bijection) F from A to B is a function that is both one-to-one and onto. 12341234 abcdeabcde A B This function is not a bijection. Bijective function

60. Definition A one-to-one correspondence (or bijection) F from A to B is a function that is both one-to-one and onto. 12341234 abcdeabcde A B Bijective function This function is not a bijection.

61. Definition A one-to-one correspondence (or bijection) F from A to B is a function that is both one-to-one and onto. 12341234 abcdabcd A B Bijective function This function is a bijection.

62. Properties Definition Let R be a binary relation on A. R is reflexive iff for all x  A, x R x. R is symmetric iff for all x,y  A, if x R y then y R x. R is transitive iff for all x,y,z  A, if x R y and y R z then x R z.

63. Equivalence relation Definition R is a equivalence relation on A iff R is a binary relation on A. R is reflexive. R is symmetric. R is transitive.

64. Transitive closure Definition Let R be a binary relation on A. The transitive closure of R is the binary relation R t on A That satisfies the following three properties: R t is transitive. R  R t. S is any other transitive that contains R then R t  S.

65. Mutually disjoint Definition Sets A 1,A 2, A 3, …,A n are Mutually disjoint (pairwise or nonoverlapping) Iff, any two sets A i,A j with distinct subscripts have not any elements in common, precisely A i  A j = empty set .

66. Set partition Definition A collection of nonempty sets {A1,A2, A3, …,An} is a Partition of a set A Iff, A = A1  A2  A3  …  An and A1,A2, A3, …,An are mutually disjoint.

67. Definition Given a partition of A={A 1,A 2, A 3, …,A n }. The binary relation induced by the partition, R, is defined on A as follows: for all x,y  A, x R y Iff, there is a subset A j of the partition such that both x and y are in A j. Set partition

68. Theorem Let A be a set with a partition and Let R be the relation induced by the partition. Then R is reflexive, symmetric and transitive. Set partition How to prove this theorem?

69. Equivalence class Definition Suppose A is a set and R is a equivalence relation on A. For each a  A, the equivalence class of a, denoted [ a ], is the set of all elements x in A such that x is related to a by R. [ a ] = { x  A | x R a }.

70. Theorem Lemma 1 Let R be an equivalence relation on A, a b  A. If a R b then [ a ] = [ b ]. Lemma 2 Let R be an equivalence relation on A, a b  A, then either [ a ]  [ b ] =  or [ a ] = [ b ]. Theorem If A is a nonempty set and R is an equivalence relation on A, then the distinct equivalence classes of R form a partition of A; that is, the union of the equivalence classes is all of A and the intersection of any two distinct classes is empty.

71. Antisymmetric Definition A relation R on a set A such that (a,b) and (b,a) are in R only if a=b, for all a,b in A, is called antisymmetric.

72. Example Let R be a relation on the set A R = { (a,b) | a < b } Findthe inverse relation R -1 and the complementary relation R.