[Math] Is $xy \geq 0$ an equivalence relation

Question

I have the following question,

Relation on the reals: $x\thicksim y$ means that $xy \geq 0$, and I have to decide whether or not it is an equivalence relation.

First, I know that it is reflexive since I just can take any positive or negative real number and I'll get: $xy \geq 0$.

Deciding whether it is symmetric, I know that if it is, then $x\thicksim y$ and $y \thicksim x $. But if it's not, then $x\thicksim y$ BUT $y \not\thicksim x$.

If I wanted to do a counterexample to show that it is not symmetric, can I just take any number x and -y? (Although I know that will mean that $x \not\thicksim y $ and $y \not\thicksim x$). I have the same problems to show if it's transitive.

Thanks in advance!

Answer 1

0 ~ -1

0 ~ 1

but not -1 ~ 1

It is not an equivalence relation