Why is $xy>0$ not an equivalence relation on the set $\mathbb{Z}$?

Question

From what I understand, an equivalence relation is a relation which must be

• reflexive
• symmetric
• transitive

The relation in question is:

• reflexive because the square of a number is always positive
• symmetric as $xy=yx$
• transitive as if $xy>0$ and $yz>0$, then $xy^2z>0$. Divide the last expression by $y^2$, and we find that $xz>0$

Answer 1

$0^2$ is not positive, so the relation is not reflexive.