For $a,b\in \mathbb Z$, let $a \sim b \iff (ab>0) \lor (a=b)$. Is $\sim$ an equivalence relation on $\mathbb Z$? If so, find the quotient set.
I've already found that $\sim$ is an equivalence relation, and I kind of get that the quotient set is a partition of the set (?) but finding the quotient set isn't really clicking with me. Could someone explain how to work through a problem like this?
This is just a more convoluted way of writing a relation you should know very well by a different description. I encourage you to show that these are in fact describing the same relation. The phrasing you should be more familiar with is:
$a\sim b$ iff $a$ and $b$ are the same sign.
Remember that positive times positive is positive and negative times negative is positive.
Reading the relation in this way it should be clear what the equivalence classes are.