Consider the set $S=[-8,8] \cap \mathbb{Z}$. Define a relation $R$ such that $(a, b) \in R$ if and only if $[a]_4 = [b]_4$.
Now the way that I understand this question is that $[x]_4$ is a remainder in the congruence class $4$. So basically $a$ and $b$ needs to have the same remainder modulo $4$.
Would it be correct to write: "Consider the relation R on all ordered in the same congruence class modulo $4$ such that $[a]_4=[b]_4$"?
If you are doing a homework problem, I would list the elements of $R$.
If you are writing an article, I prefer $$ R = \left\{\left. (a,b) \in S^2\right| a \equiv b \pmod{4}\right\}. $$