Discrete Mathematics with Applications 4th Edition

Published by Cengage Learning
ISBN 10: 0-49539-132-8
ISBN 13: 978-0-49539-132-6

Chapter 3 - The Logic of Quantified Statements - Exercise Set 3.2 - Page 117: 35


Consider for example problem 30: Statement: ∀ integers a, b, and c, if a − b is even and b − c is even, then a − c is even. Inverse: ∀ integers a, b, and c, if a − b is not even or b − c is not even, then a − c is not even. The statement is true, but its inverse is false. As a counterexample, let a = 3, b = 2, and c = 1. Then a-b=1, which is not even. But a-c=2 is even. Hence the inverse is false.

Work Step by Step

Recall that two statement forms are logically equivalent if, and only if, they have identical truth values for each possible substitution of statements for their statement variables.
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