Finding equivalence classes in $\mathbb{R} \times \mathbb{R}$

Question

We have this relation on $\mathbb{R}^* \times \mathbb{R}^*$ defined by: $(x,y) \sim (a,b)$ if $xa > 0$ and $yb>0$.

I have proven it is an equivalence relation, and I know the equivalence classes should be four: the set of all $x$ and $a$ positive, $y$ and $b$ negative; $x$, $a$, $y$ and $b$ positive; $x$ and $a$ negative, $y$ and $b$ positive; $x, a, y$ and $b$ negative.

But I'm failing to give a mathematical description of these classes, or justify it. Any help? Thank you!

Answer 1

Hint: $(1,1); (1,-1), (-1,1); (-1,-1)$ are in different classes. And every element is equivalent to one of these elements.

Answer 2

Geometrically, $x, a$ have the same sign and $y, b$ also have the same sign means the points $(x,y)$ and $(a,b)$ are in the same quadrant.