Equivalence Relation, Is [15]r = [-13]r

Question

For the equivalence relation on the integers given by $(x, y) ∈ R$ if and only if $7$ divides $x - y$, $7 | (x - y)$. Is $[15]_R = [-13]_R$? Is $[15]_R = [13]_R$ (the $R$'s would be subscripts denoting they are elements of $R$). I'm not sure where to start here, are these asking about symmetry?

Answer 1

An element $x$ is in the equivalence class ${[y]}_R$ if it satisfies the equivalence relation. $$\begin{align}x\in {[y]}_R \;& \iff\; xRy \\ & \iff\; 7\mid (y-x)\end{align}$$

Now two equivalence classes will be the same if they contain the same elements.   This means that: $${[x]}_R={[y]}_R \;\iff\; xRy$$

So we have $\;{[15]}_R={[{-}13]}_R\;$ iff $\;7\mid({-}13-15)$, and that $\;{[15]}_R={[{+}13]}_R \;$ iff $\;7\mid(13-15)$

Which is true or false?

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The truth statement, $\,m\mid n\,$, means $m$ divides $n$ with no remainder. That is, there is some integer $k$ such that $\,km = n\,$ .