Discrete Mathematics with Applications 4th Edition

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

Chapter 4 - Elementary Number Theory and Methods of Proof - Exercise Set 4.6 - Page 205: 5

Answer

Negation: "There exists a greatest even integer." Proof: There is no greatest even integer. Suppose that there is some greatest even integer $m$. Then for every even integer $k$, $k\leq m$. Now let $n=m+2$. Then $n\gt$m. But this is a contradiction, because $n$ is also an even integer. Hence, our assumption is false, and we conclude that there is no greatest even integer.

Work Step by Step

We are justified in defining $n=m+2$ by the closure of the integers under addition, and we are justified in asserting that $n\gt$m by the order-preserving property of addition. These are, however, technicalities.
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