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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.

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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 $[0]=\{0\}$).

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  • $\begingroup$ then would I have to find the partition with respect to y as well and combine them? $\endgroup$ – Pam_22R Nov 6 '16 at 16:48
  • $\begingroup$ I don't know what is expected of you (I am assuming this is a homework assignment). Personally, I would give full marks to any reasonable answer that demonstrates that the student knows how to construct the partition. $\endgroup$ – parsiad Nov 6 '16 at 16:56
  • $\begingroup$ yes its a homework assignment and it asks to give a description of what the partition would look like so I assume it wants me describe what happens both in the case of x and y So some things like P={.....} $\endgroup$ – Pam_22R Nov 6 '16 at 17:06
  • $\begingroup$ How about writing $P=\{ \{0\}, \{-1,1\}, \{-2,2\}, \ldots \}$? $\endgroup$ – parsiad Nov 6 '16 at 17:08
  • $\begingroup$ Yes, I was thinking of writing something like this but since I wants only a description I am not sure how to say this in words too well; $\endgroup$ – Pam_22R Nov 6 '16 at 17:11
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If you need to find the equivalence class for each element:

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

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