I'm having some trouble with statements of the form "if/then" vs "if and only if". Would someone mind giving me a sanity check here?
My interpretation does not seem to make a lot of sense.
Let $\,p, q , r\,$ be the propositions:
$p:\;$ You have the flu.
$q:\;$ You miss the final exam.
$r:\;$ You pass the course.
We are asked to translate the following into natural language:
$$\lnot q \leftrightarrow r$$
My interpretation: You don't miss the final exam if and only if you pass the course.
That's a correct interpretation of the desired conclusion, though it might sound a little less awkward to phrase it as follows:
"You pass the course if and only if you don't miss the final exam."
I just flipped the right-hand side and left hand side of the biconditional ("if and only statement") which is valid, since $$\begin{align} \lnot q \leftrightarrow r & \equiv (\lnot q \rightarrow r) \land (r\rightarrow \lnot q)\\ & \equiv (r\rightarrow \lnot q) \land (\lnot q \rightarrow r) \\ & \equiv r \leftrightarrow \lnot q\end{align}$$