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: 42

Answer

∃ε > 0 such that ∀ integers N, ∃ an integer n such that n > N and either L − ε ≥ $a_n$ or $a_n$ ≥ L + ε. In other words, there is a positive number ε such that for all integers N, it is possible to find an integer n that is greater than N and has the property that an does not lie between L − ε and L + ε.

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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