In "A Transition to Advanced Mathematics", the 8th edition, for chapter 1 problem 8 (c), I must explain why the following is true:
If $P$ is equivalent to $Q$, then $\sim P$ is equivalent to $\sim Q$
(Note in the chapter one we only learned about $\sim$, $\land$, $\lor$, and truth values with truth tables)
Attempt
$\sim P$ has the opposite truth value of $P$, and $P$ is equivalent to $Q$. Therefore, $\sim P$ has the opposite truth value as $Q$. Since $\sim P$ has the opposite truth value as $Q$, and $Q$ has the opposite truth value as $\sim Q$, we get $\sim P$ has the same truth value as $\sim Q$. Therefore, $\sim P$ is equivalent to $\sim Q$.
Question: Is my attempt correct? If so, how can it be improved? If not, what is the correct answer?