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 4.1 - Page 161: 1

Answer

a. Yes: $-17$ is an odd integer. b. Yes: $0$ is an even integer. c. Yes: $2k-1$ is an odd integer.

Work Step by Step

a. By definition, an integer $n$ is odd if and only if $n=2k+1$, where $k$ is some integer. Letting $k=-9$, we see that $-17=2(-9)+1$. b. By definition, an integer $n$ is even if and only if $n=2k$, where $k$ is some integer. Letting $k=0$, we see that $0=2\times0$. c. By definition, an integer $n$ is odd if and only if $n=2m+1$, where $m$ is some integer. We test to see whether $2k-1$ is odd by letting $2k-1=2m+1$, and solving for $m$ to get $m=k-1$. Thus, we see that $2k-1=2(k-1)+1$, where $k-1$ is an integer because $k$ and $1$ are integers. Therefore, $2k-1$ can be written in the form $2m+1$ where $m$ is an integer, so $2k-1$ must be odd.
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