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 $mn=1)$ $\rightarrow ((m=1 \land n=1) \lor (m=-1 \land n=-1))$'.
