Determine all equivalence classes of $xy>0$

Question

Define the equivalence relation $R$ as follows: For $x,y\in\mathbb R$, $x$ is equivalent to $y$ if and only if $xy\geq 0$. Determine all of the equivalence classes of this equivalence relation.

Answer 1

Hint : $xy \ge 0$ is $x$ and $y$ both have same sign or one among them is zero.

Answer 2

The title doesn't match the question. I'll use the wording from the question. As written in your question (not the title), $R$ is not an equivalence relation; $0$ is equivalent to $1$, and $-1$ is equivalent to $0$, but $-1$ is not equivalent to $1$, so $R$ is not transitive.

If you mean what the title says ($x$ is equivalent to $y$ if $xy > 0$), then $R$ is not an equivalence relation, but for a different reason. I'll leave it up to you to figure out why.