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 3.3 - Page 130: 37


a. Statement: ∀ even integers n, ∃ an integer k such that n = 2k. b. Negation: ∃ an even integer n such that ∀ integers k, n $\neq$ 2k. There is some even integer that is not equal to twice any other integer.

Work Step by Step

Recall the negation of a for all statement: ~($\forall x$ in D, P(x)) $\equiv \exists x$ in D such that ~P(x). Recall the negation of an exists statement: ~($\exists x$ in D, P(x)) $\equiv \forall x$ in D such that ~P(x). To negate a multiply quantified statement, apply the laws in stages moving left to right along the sentence.
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