For what $A,B \subset \mathbb R$ is $\{(x,y) \in A \times B \mid x^2-y^2=0\}$ a well defined bijection?

Question

For what $A,B \subset \mathbb R$ is relation $\{(x,y) \in A \times B \mid x^2-y^2=0\}$ a well defined bijection from $A$ to $B$?

This concept is pretty new for me so I'm not sure how to show this on the right way.

Answer 1

As it stands $x^2 - y^2 = 0$

$x = y$ or $x = -y$

So, how do we restrict $A$ and $B$ so that only one of the above holds?

Restrict $A$ to either the positive reals or the negative reals, and $B$ to either the positive reals or the negative reals.

And then for every $x\in A$ there will be a unique $y\in B$ that satisfies the relation. And for every $y \in B$ there is one $x\in A$