How to describe the partition for the given equivalence relation?

Question

The equivalence relation I am given is:

$xRy$ iff $x^2=y^2$

I am asked to describe the partition and write it out.

I started by looking at say both $x$ and $y$ can either be negative of positive. Then I concluded that $-x=y$ or $x=y$ but I am not too sure where to go from there.

Answer 1

I think you have a fine description of the partition through your assertion $$ xRy\iff x^{2}=y^{2}\iff|x|=|y|. $$ You can be more explicit by defining the partitions by $[x]=\{y\in\mathbb{R}\colon xRy\}$ for each $x\in\mathbb{R}$. That is, $[x]$ is the partition which contains $x$ and all elements equal to it, namely $-x$. That is, $[x]=\{x,-x\}$ for each $x\in\mathbb{R}$ (note that this also works for $0$, since $0=-0$ and we get $[0]=\{0\}$).

Answer 2

If you need to find the equivalence class for each element:

For $x \in \mathbb{R}$, $[x]=\{y/|y|=|x|\}$.