Implications Reading: EC 1.5 Peter J. Haas INFO 150 Fall Semester 2019 Lecture 4 1/ 19
Implications Definition and Examples The Logic of Implications Negating Implications Contrapositives, Converses, and Inverses The Language of Implication Logic Puzzles Revisited Lecture 4 2/ 19
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 Lecture 4 3/ 19
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 Definitions I Implication: A statement of the form “if p 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] Lecture 4 3/ 19
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 Definitions I Implication: A statement of the form “if p 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] Implications with predicates 1. For all real numbers x , if x 2 > 4, then x > 2. ∀ x ∈ R , ( x 2 > 4) → ( x > 2) 2. For all students s at UMass Amherst, if s is an Informatics major, then s must take 6 core courses. ∀ s ∈ U , ( x ∈ I ) → ( s must take 6 core courses) Lecture 4 3/ 19
More Implication Examples Problem: Identify the domain D , hypothesis P , and conclusion Q , so that the implication is of the form “for all x ∈ D , if P ( x ), then Q ( x )” I If a triangle has three equal sides, then it has three equal angles: D = set of all triangles, P ( t ) = “ t has three equal sides, Q ( t ) = “ t has three equal angles - set of integers I D I If an integer ends with a 2, it is a multiple of two: - a multiple of with Qin Pln ) ends is a 2 n 2 - n - - , I If a real number x has a real square root, then x is not negative: D= IR real root PCH ) has X = square a 7 ( X 2e ) 0 ) Qcx ) LX ? I = Lecture 4 4/ 19
The Logic of Implications: Example Chen Al Betty Dharmendra 19 Coke Beer 25 Example: I A trooper walks into a pub: Al, Betty, Chen, and Darmendra are drinking I Bartender says everyone is obeying the law 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 , and Q to form the implication: “for all x ∈ D , if P then Q ” students D= is drinking in pub RX ) beer x - - , Q Cx ) at least Years of X is = 21 age 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? Needs to turn Al and Chen over need to check f P ( Betty ) age - so no - matter what he drinks ok Dharmendra ) T no so Q C = Lecture 4 5/ 19
The Logic of Implications Chen Al Betty 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 p → q p q T T T T F F F T T Betty { F F T Lecture 4 6/ 19
More Examples n 3 − n Divisible by 4? n @ 1 0 0=4×0 ✓ 3 24 if 24 4×6 ' 5 Problem: For every positive integer n , is if n is odd, then n 3 − n is divisible by 4 7 I Hypothesis: :* :÷÷÷i::÷ "÷÷ . I Conclusion: I Is the statement true or false? n 3 − n Divisible by 4? n NO 2 6 Problem: For all integers n , Yes if 3 n = 9, then n 2 = 9 4 Go NO 6 40 I Why is the above statement true? Yes 8 counterexample 504 no Problem: For all integers n , if n 2 > 9, then n > 3 Concl. ( n 2 = 9) n Hyp. (3 n = 9) I Why is the above statement false? IF T -3 F - 4 counterexample is a 0 p , 3 T 10 f f Lecture 4 7/ 19
Summary Rule 1 For a statement of the form “if hypothesis, then conclusion” to be false, it must be the case that the hypothesis is true and the conclusion is false Rule 2 For a quantified statement of the form “ ∀ x , P ( x ) → Q ( x )” to be false, it must be the case that at least one value of x is a counterexample such that P ( x ) is true but Q ( x ) is false. Lecture 4 8/ 19
Negating Implications Proposition 1 The negation of p → q is p ∧ ¬ q Proof: ¬ ( 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 ∀ x ∈ D , P ( x ) → Q ( x ) is ∃ x ∈ D , P ( x ) ∧ ¬ Q ( x ) Proof: � ∀ x ∈ D , P ( x ) → Q ( x ) � = ∃ x ∈ D , ¬ � P ( x ) → Q ( x ) � = ∃ x ∈ D , P ( x ) ∧ ¬ Q ( x ) ¬ Lecture 4 9/ 19
Negating Implications: Examples Problem: Negate each of the following statements: I If Bob has an 8:00 class today, then it is Tuesday , but today is not Tuesday a e class today Bob has 8 : an I If Juanita gets chocolate, then she has a happy birthday , but has chocolate unhappy Juanita gets an birthday I For all real numbers x , if x > 2, then x 2 > 4 said that x 's 4 but there exist xc.IR X L > , - Lecture 4 10/ 19
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): T I P ( n ) → Q ( n ): “If n ends in the digit 2, then n is divisible by 2” [ F I Q ( n ) → P ( n ): “If n is divisible by 2, then n ends in the digit 2” C f I ¬ P ( n ) → ¬ Q ( n ): “If n does not end in the digit 2, then n is not divisible by 2” T I ¬ Q ( n ) → ¬ P ( n ): “If n is not divisible by 2, then n does not end in the digit 2” Definition: for the implication ∀ x ∈ D , P ( x ) → Q ( x ): I The converse is ∀ x ∈ D , Q ( x ) → P ( x ) I The inverse is ∀ x ∈ D , ¬ P ( x ) → ¬ Q ( x ) I The contrapositive is ∀ x ∈ D , ¬ Q ( x ) → ¬ P ( x ) Analogous definitions for proposition p → q : I Converse: q → p I Inverse: ¬ p → ¬ q I Contrapositive: ¬ q → ¬ p Lecture 4 11/ 19
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 inverse converse counterexample H b - D= 2 , converse odd ) Example: Lgb even ) are Cath is → I A true implication whose converse is false: ( a and b are odd) → ( a + b is even) I A true implication whose converse is true: ( n is even) → ( n 2 is even) Lecture 4 12/ 19
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 0 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 Lecture 4 13/ 19
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 ∀ x ∈ D , Q ( x ) or ∀ x ∈ U , x ∈ D → Q ( x ) (The latter is preferable for proof-writing) Lecture 4 13/ 19
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 ∀ x ∈ D , Q ( x ) or ∀ x ∈ U , x ∈ D → Q ( x ) (The latter is preferable for proof-writing) Example: D = set of UMass Informatics students & U = set of UMass students I Statement: For all s ∈ D , s must take discrete math I Implication: For all s ∈ U , if s 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 4 13/ 19
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