Implications. Reading: EC 1.5. Peter J. Haas. INFO 150 Fall Semester Lecture 5 1/ 17

Transcription

1 Implications Reading: EC 1.5 Peter J. Haas INFO 150 Fall Semester 2018 Lecture 5 1/ 17

2 Implications Definition and Examples The Logic of Implications Negating Implications Contrapositives, Converses, and Inverses The Language of Implication Logic Puzzles Revisited Lecture 5 2/ 17

3 Implications Informal examples 1. If I am voting at a polling place, then there is an election today 2. If it is snowing, then the streets are slippery 3. If you are an Informatics major, then you must take 6 core courses 4. If a real number x satisfies x 2 > 4, then x > 2 ] ] propositions predicates Chatwinbled Definitions I Implication: Astatementoftheform ifp is true, then q is true I Notation: p! q means p implies q [! has lower precedence than ^, _, ] I p is the hypothesis and q is the conclusion [can be propositions or predicates] A more precise version of implications with predicates 3. For all students s at UMass Amherst, if s is an Informatics major, then s must take 6 core courses. 4. For all real numbers x, ifx 2 > 4, then x > 2. Lecture 5 3/ 17

4 - More Implication Examples Problem: Identify the domain D, hypothesisp, andconclusionq, sothatthe implication is of the form for all x 2 D, ifp(x), then Q(x) I If a triangle has three equal sides, then it has three equal angles: D =setofalltriangles,p(t) = t has three equal sides, Q(t) = t has three equal angles I If an integer ends with a 2, it is a multiple of two: D " I Pln ) ends with - n an Qcnj I If a quadrilateral can be inscribed in a circle, then the opposite angles of the quadrilateral sum to 180 : in a circle in - x of quadrilaterals, Play) is a multiple can be inscribed of 2 -- D set QLD ) opposite angles of X Sum to 1800 = I If a real number x has a real square root, then x is not negative: a D= R PC xx x has real square QCX) x so ' root Lecture 5 4/ 17

5 -. The Logic of Implications: Example Example: I Atrooperwalksintoapub 19 Y Coke N Beer Y 25 N I Al, Betty, Chen, and Darmendra are drinking beverages I Bartender says everyone is obeying the law Al Betty Chen Dharmendra I The law: if you are drinking beer, then you are at least 21 years of age I In front of each person is a card with age on one side and beverage on the other Problem: I Identify D, P, andq to form the implication: for all x 2 D, ifp then Q - s is drinking beer D= students in pub Pls ) Qbs ) = s is at least u yrs I Whose cards does the trooper need to turn over to check that everyone is obeying the law? Why does she not need to turn over the other cards? about Need to look at Al & Chen only cane ) No need to look at Betty c pls ) is false where ) cases e No need to look at Dharmendra C QLD is true ) pls, isatysgeisanpdabe Lecture 5 5/ 17

6 The Logic of Implications Al Betty Chen Dharmendra 19 Coke Beer 25 If you are drinking beer, then you are at least 21 Example summary: The only time that p! q is false is if p is true and q is false Truth table for implication Betty { p q p! q T T T T F F F T T F F T Lecture 5 6/ 17

7 More Examples n n 3 n Divisible by 4? 1 o Yes 0--4,0 ' 3 24 Yes 6 Problem: For every positive n'unrsdiv.by - 30 integer n, Yes 120=4 if n is odd, then n 3 n is divisible Yes 336=4-84 I Hypothesis: n is odd I Conclusion: 4 24*4 I Is the statement true or false? I n n 3 n Divisible by 4? irrelevant to Problem: For all integers n, 2 No if 3n =9,thenn 2 60 =9 4 Yes truth of implication : NO I Why is the above statement true? ninfoodd yes Problem: For all integers n, if n 2 > 9, then n > 3 n Hyp. (3n =9) Concl. (n 2 =9) I Why is the above statement false? -3 F T 0 counterexample! f, 3 T T 10 n= -4 F F Lecture 5 7/ 17

8 - I Summary Rule 1 For a statement of the form if hypothesis, then conclusion to be false, itmustbe the case that the hypothesis is true and the conclusion is false Rule 2 For a quantified statement of the form 8x, P(x)! Q(x) to be false, itmustbethe case that at least one value of x is a counterexample such that P(x) istruebut Q(x) is false. Problem: Determine whether each of the following statements is true or false (give a counterexample if false) I 8x 2 R, if x 2 5x +4=0,thenx > 0 I 8n 2 Z, ifn 2 =1,thenn 3 =1 f ne qx-tbxtc-0-bt.fi/ac I 8a, b 2 Z >0,ifa and b are both odd, then a + b is also odd I 8a, b 2 Z >0,ifa and b are both odd, then ab is also odd T T I 's a counterexample -3--4,1 Ta Straggly 'll Fasb a 7 b a 5 z Lecture 5 8/ 17

9 Negating Implications Proposition 1 The negation of p! q is p ^ q Proof: a d p I p q p! q (p! q) q p ^ q - T T T F F F T F F T T T F T T F F F F F T F T F Note: The negation of an implication is not an implication! Proposition 2 The negation of 8x 2 D, P(x)! Q(x) is 9x 2 D, P(x) ^ Q(x) Proof: 8x 2 D, P(x)! Q(x) = 9x 2 D, P(x)! Q(x) = 9x 2 D, P(x) ^ Q(x) Lecture 5 9/ 17

10 Negating Implications: Examples Problem: Negate each of the following statements: I If Bob has an 8:00 class today, then it is Tuesday I If Juanita gets chocolate, then she has a happy birthday I For all real numbers x, ifx > 2, then x 2 > 4 There exists a real number X seṭ > 2 and I X 254 f I For all real numbers x > 0, if x 2 =1,thenx 3 =1T Problem: Negate each of the following statements: I If you buy the extended warranty, then nothing will go wrong with your ipad I If Chris gets a flu shot, then he will not get the flu I For all triangles t, ift has three equal sides, then t has three equal angles I 8x 2 {1, 2, 3, 4, 5}, x 2 is positive ^ n ( p q ) =p a Tq ft x. P the Quy )) x, Bob has an 8:00 class today and it is not Pindi Tuesday Juanita gets chocolate and she Has an unhappy birthday There exist a real number x so with The I such that k 1 E) You buy extended Warranty and something goes wrong " IPad Chris gets flu a shot and he gets the flu There exists a triangle f such that t has three e sides There exists xe 5143,4, I X E 31,43 4,, s }, X so s } such that I is negative - but not thence = angles Lecture 5 10/ 17

11 Contrapositives, Converses, and Inverses Example: P(n) = n ends in the digit 2 and Q(n) = n is divisible by 2 True or False (if False give a counterexample): I P(n)! Q(n): Ifn ends in the digit 2, then n is divisible by 2. T I Q(n)! P(n): Ifn is divisible by 2, then n ends in the digit 2 F I P(n)! Q(n): Ifn does not end in the digit 2, then n is not divisible by 2 f I Q(n)! P(n): Ifn is not divisible by 2, then n does not end in the digit 2, T Definition: for the implication 8x 2 D, P(x)! Q(x): I The converse is 8x 2 D, Q(x)! P(x) I The inverse is 8x 2 D, P(x)! Q(x) I The contrapositive is 8x 2 D, Q(x)! P(x) Analogous definitions for proposition p! q: I Converse: q! p I Inverse: p! q I Contrapositive: q! p Lecture 5 11/ 17

12 . More on Contrapositives, Converses, and Inverses Proposition 1. An implication and its contrapositive are logically equivalent 2. The converse and inverse of an implication are logically equivalent 3. An implication is not logically equivalent to its converse (nor its inverse) implication contrapositive no relation in general converse Example: I Atrueimplicationwhoseconverseisfalse: (a and b are odd)! (a + b is even) I Atrueimplicationwhoseconverseistrue: (n is even)! (n 2 is even) inverse Counterexamples La a and at b even ( ). r, b -- 4 bare odd ) is even Cnn ) I n is even ) Lecture 5 12/ 17

13 The Language of Implication Don t confuse sentence rearrangement with logical converse I Statement: If an integer m ends in the digit 0, then m is a multiple of 5 I Same statement: An integer m is a multiple of 5 if it ends in the digit 0 I Converse: If an integer m is a multiple of 5, then m ends in the digit 0 For all statements can be written as an implication or not I Suppose D is a subset of a larger set U I Then we can write 8x 2 D, Q(x) or 8x 2 U, x 2 D! Q(x) (The latter is preferable for proof-writing) n ( peg ) = p n 7g Example: D =setofumassinformaticsstudents&u =setofumassstudents I Statement: For all s 2 D, s must take discrete math I Implication: For all s 2 U, ifs is an Informatics student, then s must take discrete math I Negation of statement: there exists an Informatics student who does not have to take discrete math I Negation of implication: There exists a UMass student who is an Informatics student but does not have to take discrete math Lecture 5 13/ 17

14 Industrial-Strength Predicates Example I Domain D =setofallcars I For any red sports car, you can find an expensive car that is safer I S(x) = x is a sports car I E(x) = x is expensive I R(x) = x is red I A(x, y) = x is safer than y I 8x 2 D, S(x) ^ R(x)! 9y 2 D, E(y) ^ A(y, x) I ) I Negation: 9x 2 D, S(x) ^ R(x) ^ 8y 2 D, E(y) _ A(y, x) air :D ix. I There exists a red sports car such that any car is either not more expensive or is less safe " g) ( p x, = pig p n - ol Lecture 5 18/ 18