Implication (6A) Young Won Lim 3/17/18

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1 Implication (6A) 3/17/18

2 Copyright (c) Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled "GNU Free Documentation License". Please send corrections (or suggestions) to youngwlim@hotmail.com. This document was produced by using LibreOffice and Octave. 3/17/18

3 Material Implication & Logical Implication Given two propositions A and B, If A B is a tautology It is said that A logically implies B (A B) in every interpretation Material Implication A B Logical Implication A B (not a tautology) (a tautology) A B A B A B A B T T T T F F F T T F F T A B A B A B A T T T T T F F T F T F T F F F T tautology A B A Logic (6A) Implication 4 3/17/18

4 Logic and Venn diagram (1) case case case case S N S N T T T T F F F T T F F T S N Logic (6A) Implication 12 3/17/18

5 Logic and Venn diagram (2) p r q case case case case case case case case p q r p q T T T T T T F T T F T F T F F F F T T T F T F T F F T T F F F T Logic (6A) Implication 13 3/17/18

6 Material Implication and Venn Diagram case case case case S N S N T T T T F F F T T F F T S N When S N is True S N S N S N Logic (6A) Implication 14 3/17/18

7 When S N is a true statement case case case case S N S N T T T T F F F T T F F T (1) (2) S N S N is True cases ++ (1) (2) if the conditional statement (S N) is a true statement, then the consequent N must be true if S is true the antecedent S can not be true without N being true Logic (6A) Implication 15 3/17/18

8 S N if the conditional statement (S N) is a true statement, then the consequent N must be true if S is true S N S N case cases + S N x S x N Logic (6A) Implication 16 3/17/18

9 ~N ~S if the conditional statement (S N) is a true statement, the antecedent S can not be true without N being true N S N N S case cases + N S x N x S Logic (6A) Implication 17 3/17/18

10 Material Implication vs. Logical Consequence Material Implication Logical Consequences S N S N S N N S T F exists Always True (Tautology) entailment S N S N syntactic (proof) semantic (model) Logic (6A) Implication 18 3/17/18

11 Implication S N S N If S, then N. S implies N. N whenever S. S is sufficient for N. S only if N. not S if not N. not S without N. N is necessary S. Logic (6A) Implication 19 3/17/18

12 Necessity and Sufficiency S N condition that guarantees N sufficiency for N S satisfies at least N N if S condition that must be satisfied for S necessity for S without N, it can t be S S only if N N S Logic (6A) Implication 20 3/17/18

13 Other Necessary Conditions requirement 2 S N Get A Term Paper submitted Midterm taken Get A Term Paper Final test taken Rank < 15% necessary conditions of S requirement 1 requirement n S only if N not S if not N Logic (6A) Implication 21 3/17/18

14 Other Sufficient Conditions S N Case 2 score 100 on every work Get A Case 1 Get A score 100 on every work Case 3 case 1: 100, 95, 90, case 2: 80, 95, 99, Case n sufficient conditions of N Term Paper necessary conditions of Get A N if S N whenever S Implication (2A) 22 3/17/18

15 Resolution (7A) 3/19/18

16 Copyright (c) Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled "GNU Free Documentation License". Please send corrections (or suggestions) to youngwlim@hotmail.com. This document was produced by using LibreOffice and Octave. 3/19/18

17 Argument ( p q) ( p r) q r ( p q) ( p r) q r Logic (7A) Resolution 3 3/19/18

18 Truth Table p q r p q T T T T T T F T T F T T T F F T F T T T F T F T F F T F F F F F p q r p p r T T T F T T T F F F T F T F T T F F F F F T T T T F T F T T F F T T T F F F T T p q r ( p q) ( p r) q r T T T T T T T F F T T F T T T T F F F F F T T T T F T F T T F F T F T F F F F F ( p q) ( p r) q r Logic (7A) Resolution 4 3/19/18

19 Interpretation of this truth table case case case case case case case case A B p q r p q p r ( p q) ( p r) q r T T T T T T T T T F T F F T T F T T T T T T F F T F F F F T T T T T T F T F T T T T F F T F T F T F F F F T F F A B T T T T T T T T Whenever p q and p r are true, q r is true ( p q) ( p r) q r Logic (7A) Resolution 5 3/19/18

20 Venn diagram for p q p q case case case case case case case case p q r p q T T T T T T F T T F T T T F F T F T T T F T F T F F T F F F F F Logic (7A) Resolution 6 3/19/18

21 Venn diagram for p r p r case case case case case case case case p q r p p q T T T F T T T F F F T F T F T T F F F F F T T T T F T F T T F F T T T F F F T T Logic (7A) Resolution 7 3/19/18

22 When ( p q) ( p r) is true p q p r When p q and p r is true is true ( p q) ( p r) cases +++ Logic (7A) Resolution 8 3/19/18

23 When ( p q) ( p r) is true, q r is also true p q p r q r cases +++ ( p q) ( p r) q r cases Logic (7A) Resolution 9 3/19/18

24 Argument Case 1: p is false Case 2: p is true p q p r q r F q T r q T q F r r when p is false, q must be true. when p is true, r must be true. Therefore regardless of truth value of p, If both premises hold, then the conclusion q r is true Logic (7A) Resolution 10 3/19/18

25 Resolution Examples p q p r q r p q p r p q p r q r p p q q p p q p p q q p p r r p p r p p r r Logic (7A) Resolution 11 3/19/18

26 Resolution in Prolog Conjunctive Norm Form (CNF) is assumed ( ) ( ) ( ) the variables A, B, C, D, and E are in conjunctive normal form: A ( B C ) ( A B ) ( B C D ) ( D E ) A B A B The following formulas are not in conjunctive normal form: ( B C ) ( A B ) C A ( B ( D E ) ) Logic (7A) Resolution 12 3/19/18

27 Example A p q p r p q q r q p r q r p r r p ( p q) ( p r) ( p q) ( q r) ( p q) ( p r) ( p q) ( q r) (q) ( p q) ( p r) ( p q) ( q r) (q) (r) ( p q) ( p r) ( p q) ( q r) (q) (r) ( p) James Aspnes, Notes on Discrete Mathematics, CS 202: Fall 2013 Logic (7A) Resolution 15 3/19/18

28 Example B (1) p q r p q r q p q r p q r q q q r r q r q q q r = T r = T p q r p q r q p q r p q r q p r Discrete Mathematics, Johnsonbough Logic (7A) Resolution 16 3/19/18

29 Example B (2) p q r p q r q p q r p q q r p q r q p q r p r Discrete Mathematics, Johnsonbough Logic (7A) Resolution 17 3/19/18

30 Truth Table p q r p p q r T T T F T T T F F T T F T F T T F F F F F T T T T F T F T T F F T T T F F F T T p q r q p q T T T F T T T F F T T F T T T T F F T T F T T F F F T F F F F F T T T F F F T T p q r q r T T T T T T F T T F T T T F F F F T T T F T F T F F T T F F F F p q r p q r p q q r H 1 H 2 H 3 p r T T T T T T T T T T F T T T T T T F T T T T T T T F F F T F F T F T T T F T F T F T F T F T F F F F T T T T T T F F F T T F F F H 1 = p q r H 2 = p q H 3 = q r H 1 H 2 H 3 H 3 H 1 H 2 H 3 H 2 H 1 H 2 H 3 H 1 H 1 H 2 H 3 (p r) Logic (7A) Resolution 18 3/19/18

31 References [1] [2] 3/19/18

32 Functions (4A) 3/16/18

33 Copyright (c) 2015 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled "GNU Free Documentation License". Please send corrections (or suggestions) to youngwlim@hotmail.com. This document was produced by using OpenOffice and Octave. 3/16/18

34 Function A function f takes an input x, and returns a single output f(x). One metaphor describes the function as a "machine" or "black box" that for each input returns a corresponding output. A function that associates any of the four colored shapes to its color. Functions (4A) 3 3/16/18

35 Function The above diagram represents a function with domain {1, 2, 3}, codomain {A, B, C, D} and set of ordered pairs {(1,D), (2,C), (3,C)}. The image is {C,D}. However, this second diagram does not represent a function. One reason is that 2 is the first element in more than one ordered pair. In particular, (2, B) and (2, C) are both elements of the set of ordered pairs. Another reason, sufficient by itself, is that 3 is not the first element (input) for any ordered pair. A third reason, likewise, is that 4 is not the first element of any ordered pair. Functions (4A) 4 3/16/18

36 Domain, Codomain, and Range X : domain Y : domain range Functions (4A) 5 3/16/18

37 Function condition : One emanating arrow Each, one starting arrow Some, no starting arrow Some, two starting arrows Function (O) Function (X) Function (X) Relation (O) Relation (O) Relation (O) Functions (4A) 6 3/16/18

38 Composite Function A composite function g(f(x)) can be visualized as the combination of two "machines". The first takes input x and outputs f(x). The second takes as input the value f(x) and outputs g(f(x)). Functions (4A) 7 3/16/18

39 Injective Function An injective nonsurjective function (injection, not a bijection) An injective surjective function (bijection) A non-injective surjective function (surjection, not a bijection) Functions (4A) 8 3/16/18

40 Injective Function In mathematics, an injective function or injection or one-to-one function is a function that preserves distinctness: it never maps distinct elements of its domain to the same element of its codomain. every element of the function's codomain is the image of at most one element of its domain. The term one-to-one function must not be confused with one-to-one correspondence (a.k.a. bijective function), which uniquely maps all elements in both domain and codomain to each other. Functions (4A) 9 3/16/18

41 Surjective Function A surjective function from domain X to codomain Y. The function is surjective because every point in the codomain is the value of f(x) for at least one point x in the domain. A non-surjective function from domain X to codomain Y. The smaller oval inside Y is the image (also called range) of f. This function is not surjective, because the image does not fill the whole codomain. In other words, Y is colored in a two-step process: First, for every x in X, the point f(x) is colored yellow; Second, all the rest of the points in Y, that are not yellow, are colored blue. The function f is surjective only if there are no blue points. Functions (4A) 10 3/16/18

42 Surjective Functions Range = Codomain Every, arriving arrow(s) Surjective (O) Surjective (X) Surjective (X) Functions (4A) 11 3/16/18

43 Injective Functions Every, Less than one arriving arrow Injective (O) Injective (O) Injective (X) Functions (4A) 12 3/16/18

44 Surjective Function A surjective function is a function whose image is equal to its codomain. a function f with domain X and codomain Y is surjective if for every y in Y there exists at least one x in X with f(x) = y. Functions (4A) 13 3/16/18

45 Surjective Functions Surjective composition: the first function need not be surjective. Another surjective function. (This one happens to be a bijection) A non-surjective function. (This one happens to be an injection) Functions (4A) 14 3/16/18

46 Types of Functions injection surjection bijection Functions (4A) 15 3/16/18

47 References [1] [2] 3/16/18

48 Set Operations (15A) 3/5/18

49 Copyright (c) Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled "GNU Free Documentation License". Please send corrections (or suggestions) to youngwlim@hotmail.com. This document was produced by using LibreOffice and Octave. 3/5/18

50 Unions Set Operations (15A) 3 3/5/18

51 Properties of Unions Set Operations (15A) 4 3/5/18

52 Intersections Set Operations (15A) 5 3/5/18

53 Properties of Intersections Set Operations (15A) 6 3/5/18

54 Complements Set Operations (15A) 7 3/5/18

55 Complements Set Operations (15A) 8 3/5/18

56 Complements Set Operations (15A) 9 3/5/18

57 Inclusion and Exclusion Set Operations (15A) 11 3/5/18

58 De Morgan s Law Set Operations (15A) 13 3/5/18

59 The Same Cardinality Set Operations (15A) 14 3/5/18

60 References [1] [2] 3/5/18

61 Matrix (2A) 3/13/18

62 Copyright (c) Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled "GNU Free Documentation License". Please send corrections (or suggestions) to youngwlim@hotmail.com. This document was produced by using LibreOffice and Octave. 3/13/18

63 Row Vector, Column Vector, Square Matrix Functions (4A) 3 3/13/18

64 Matrix Notation Functions (4A) 4 3/13/18

65 Matrix Addition Functions (4A) 5 3/13/18

66 Scalar Multiplication Functions (4A) 6 3/13/18

67 Transposition Functions (4A) 7 3/13/18

68 Matrix Multiplication Functions (4A) 8 3/13/18

69 Properties Functions (4A) 9 3/13/18

70 Matrix Types Functions (4A) 10 3/13/18

71 Identity Matrix Functions (4A) 11 3/13/18

72 Inverse Matrix Functions (4A) 12 3/13/18

73 Inverse Matrix Examples Functions (4A) 13 3/13/18

74 Solving Linear Equations A set of linear equations If the inverse matrix exists a x + b y = e [ a b d][ x ] c y = [ e ] [ x ] c x + d y = f f y = [ a b c d] 1[ e f ] a d b c = ad bc 0 [ x y ] = 1 [ d b ][ e ] ad bc c a f e f d b = de bf [ x y ] = 1 ad bc [ de bf ce+af ] a c e f = af ce Cramer's Rule x = e f a c d b b d y = a c a c e f b d Functions (4A) 14 3/13/18

75 Rule of Sarrus (1) Determinant of order 3 only [a11 a1 2 a13 a 21 a 2 2 a 23 a 31 a 3 2 a 33] [a1 1 a12 a13 a 2 1 a 22 a 23 a11 a12 a 3 1 a 32 a 33] a 21 a 2 2 a 31 a 3 2 Rule of Sarrus [a1 1 a12 a1 a1 3 a 2 1 a 22 a 2 3 a 3 1 a 32 a 3 3] 1 a12 a 2 1 a 22 a 3 1 a 32 Copy and concatenate [a1 1 a12 a1 a1 3 a 2 1 a 22 a 2 3 a 3 1 a 32 a 3 3] 1 a12 a 2 1 a 22 a 3 1 a 32 + a 11 a 22 a 33 + a 1 2 a 2 3 a a 13 a 21 a 3 2 a 13 a 22 a 3 1 a 1 1 a 2 3 a 32 a 12 a 2 1 a 3 3 Determinant (3A) 15 03/13/2018

76 Linear Equations (Eq 1) (Eq 2) a 11 x 1 + a 1 2 x a 1 n x n = b 1 a 2 1 x 1 + a 22 x a 2n x n = b 2 (Eq 3) a n1 x 1 + a n 2 x a n n x n = b n a 11 a 1 2 a 1 n x 1 = b 1 a 2 1 a 22 a 2n x 2 = b 2 A x = b a n1 a n 2 a n n x n = b n Determinant (3A) 16 03/13/2018

77 Cramer's Rule (1) solutions A a 11 a 1 2 a 1 n a 2 1 a 22 a 2n a n1 a n2 a n n x x 1 x 2 x n b = b 1 = b 2 = b n A 1 ab 11 1 a 1 2 a 1 n ab 21 a 2 2 a 2 n a 11 ab 1 2 a 1 n a 2 1 a 2 2 a 2 n b 2 A 2 A n a 11 a 1 2 ba 1 n a 2 1 a 2 2 ba 22n ab m1 n a n 2 a n n a n 1 a m2 a n n b n a n 1 a n 2 ba nmn x 1 = det( A 1) det( A) x 2 = det( A 2 ) det( A) x n = det( A n ) det( A) Determinant (3A) 17 03/13/2018

78 References [1] [2] 3/13/18

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