I think the wording is throwing me off, and I also haven't done math in 4 months so basically my mind is scrambled eggs.
Let $\sim$ be a relation on $\Bbb Z$ defined by letting $m \sim n$ if $mn>0$. Is $\sim$ an equivalence relation?
I would say no, as $m$ or $n$ could be $0$, because $0$ is an element of $\Bbb Z$ and the union of the cells of the partition must equal the original set, but the union doesn't include $0$. Am I right? The word "if" for some reason is boggling my mind.
You are correct: It is not true that $0 \sim 0$, since $0 \cdot 0 = 0$. So the relation is not reflexive.