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

To show it is (not) symmetric you must show that $x$ ~ $y \Rightarrow y $~$ x $ or not.

If $x $~$ y $ is defined as $xy \geq 0$, does $xy \geq 0 \Rightarrow yx \geq 0$?

For the transitivity, you have to check if $x $~$ y \wedge y $~$ z \Rightarrow x $~$ z \iff xy \geq 0 \wedge yz \geq 0 \Rightarrow xz \geq 0$

Answer 2

0 ~ -1

0 ~ 1

but not -1 ~ 1

It is not an equivalence relation

Answer 3

However, note the relation $x\mathcal R y\stackrel{\text{def}}{\iff}xy>0\;$ is an equivalence relation on $\mathbf R^*$, and it has $2$ classes: positive numbers and negative numbers.