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D isc r e t e

s tructu r e s. D isc r e t e. 2110200. 2007. The foundations C ounting theory N umber theory G raphs & trees. Evaluation. The foundations 15 % Counting theory 15 % Number theory 15 % Graphs & trees 15 % Final 40 %. Motivation. How fast can we do the computation ?. Motivation.

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D isc r e t e

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  1. structures Discrete 2110200 2007 The foundations Counting theory Number theory Graphs & trees

  2. Evaluation • The foundations 15 % • Counting theory 15 % • Number theory 15 % • Graphs & trees 15 % • Final 40 %

  3. Motivation How fast can we do the computation ?

  4. Motivation ?

  5. Motivation K = 0 FOR I1 = 0 TO M { FOR I2 = 0 TO I1 { FOR I3 = 0 TO I2 { … FOR In = 0 TO In-1 { K = K + 1 } … } } } Find the value of K

  6. s s T T he Foundation Relational structures

  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. Summary Theorem : 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pp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p pCp • Domination laws:(Universal bound laws) pTT pCC • Idempotent laws: ppp ppp • Negation laws: ppT ppC • Double negative laws:  (p) p • De Morgan’s laws:  (pq) pq •  (pq) pq • Absorption laws: p(pq) pp(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 nth 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 99th statement is true.

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

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

  16. Conditional statement p q 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.

  17. Valid arguments 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) An argument is valid means that if all hypotheses are true, the conclusion is also true.

  18. Valid arguments VALID Example Given an argumentp (q  r);  r ;  p  q TRUE ( p1 p2  p3  …  pn)  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 • Disjunctive addition • Conjunctive simplification • Conjunction addition • Modus ponens • Modus tollens • Hypothetical syllogism • Disjunctive syllogism • Resolution • Dilemma P  Q  P

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

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

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

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

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

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

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

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

  29. Example Given two logical operators, p | q means  ( p  q ) p  q means  ( 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 Consider the following statements: All students go to school. John is a student. Diana is a student. …… Of 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. Universal Modus Ponens Consider a statement x P(x)  Q(x). For a particular e, P(e) is true, therefore Q(e) is true.

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

  35. Rules of inferences 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)

  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. Example Express the following theorem using the first order predicate logic. 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 membersof 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 • L1={ n| for n = 123 … } • L2={ n | for n = 1357 … } • L3={ n | for n = 14916 … } • L4={ n | for n =34822 … }. Membership problem

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

  43. Example Let U be the universe described by U = { x | 1000  x  9999}. Let Ai be the set of all numbers in U such that the ith position is i. ? Find the cardinality of the union of A1 A2 A3 and A4

  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 Union Mutually disjoint Intersection Partition Different Disjoint Complement Power set

  46. Set • Theorem Given sets A,B and C. • Commutative laws: AB = BA AB = BA • Associative laws:(AB)C = A(BC) • Distributive laws: A(BC) = (AB)(AC) • Idempotent laws: AU = A AU = U • De Morgan’s laws: (AB)c = Ac Bc (AB)c = Ac Bc • Alternative representation for set difference A-B = ABc • 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 ) Given A  C = B  C Must it be the case that A = B ? YES

  49. Multisets Definition Multisets are unordered collections of elements where an element can occur as a member more than once. { m1.a1, m2.a2, m3.a3, …, mr.ar } mi are called the multiplicities of the elements ai. OPERATORS: UNION, INTERSECTION, DIFFERENCE, SUM

  50. Multisets Fuzzy sets Definition Multisets are unordered collections of elements where an element can occur as a member more than once. { m1.a1, m2.a2, m3.a3, …, mr.ar } mi are called the multiplicities of the elements ai. 0  mi  1 Degree of membership OPERATORS: UNION, INTERSECTION, DIFFERENCE, SUM

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