Discrete Mathematics and Its Applications, Seventh Edition

Published by McGraw-Hill Education
ISBN 10: 0073383090
ISBN 13: 978-0-07338-309-5

Chapter 1 - Section 1.7 - Introduction to Proofs - Exercises - Page 91: 29

Answer

See the solution.

Work Step by Step

This statement is true, so we will prove it. Note that we must prove the conditional sentence '$((m$ and $n$ are integers with $mn=1)$ $\rightarrow ((m=1 \land n=1) \lor (m=-1 \land n=-1))$'. $Proof.$ Suppose $m$ and $n$ are integers, and suppose $mn=1$. We must show either both $m$ and $n$ are $1$ or both $m$ and $n$ are $-1$. Since $mn\neq 0$, we know $m\neq 0$ and $n\neq 0$ (this is the contrapositive of 'if $m=0$ or $n=0$, then $mn=0$', which is certainly true). Thus since $m\neq 0$, we may divide both sides of $mn=1$ by $m$ to get $n=\frac{1}{m}$. Since $n$ is an integer, we must have either $m=1$ or $m=-1$ (otherwise, $n$ would not be an integer). Now, if $m=1$, then $n=\frac{1}{1}=1$, and if $m=-1$, then $n=\frac{1}{-1}=-1$. Hence either $m=1$ and $n=1$ or $m=-1$ and n=$-1$. Therefore, '$((m$ and $n$ are integers with $mn=1)$ $\rightarrow ((m=1 \land n=1) \lor (m=-1 \land n=-1))$'.
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.