Determining all distinct equivalence classes Let
A = {−6, −5, −4, −3, −2, −1, 0, 1, 2, 3, 4}
and define a relation R on A as follows:
For all
m, n is in Z,
$m R n ⇔ 3|(m^2 − n^2)$
It is a fact that R is an equivalence relation on A. Use set-roster notation to list the distinct equivalence classes of R.
I believe I can go through each element and find all the pairs that satisfy $3|m^2-n^2$, however, this seems like it will take a while. I know I just have to find $3|(m-n)$ or $3|(m+n)$, but this still doesn't seem like the best method. What is the most efficient way of doing this?

Your goal is to find representatives of each equivalence class. Write $m=3k+r$ where $r\in\{0,1,2\}$. If $r=0$, then $n\equiv 0\pmod{3}$ is equivalent to $m$. So the first representative is $\{r=0\}$.
If $r=1$, then $n\equiv 1,2\pmod{3}$ are equivalent to $m$, so the next representative is $\{r=1\}$.
Since equivalence is reflexive, we don't need to consider $r=2$.
