Proving a relation is an equivalence relation specifically proving transitivity

Question

I'm currently studying for an exam and I've come across this question:

• Define a relation R on Z by

  xRy ⇔ 6|($x^{2} − y^{2}$)

  for x, y ∈ Z.

Prove that R is an equivalence relation and describe the equivalence classes of R

I understand how to prove it's reflexive, and I've tried to prove it's symmetric but I used the fact that -6|($y^{2}-x^{2}$) which doesn't seem like the correct way to answer this question, and I have no idea how to prove it's transitive, any help would be greatly appreciated. Thanks in advance!

Answer 1

Your proof that it's symmetric is almost certainly correct (but if you want to post the details, I could critique it).

To prove transitivity, assume $x \sim y$ and $y\sim z$.

By definition, $6| (x^2-y^2)$ and $6|(y^2-z^2)$. Another way of putting this would be to say there are integers $m,n$ such that

$x^2-y^2 = 6m$

and

$y^2-z^2 = 6n$.

If we just add those two equations together, we get

$x^2 - z^2 = 6(m+n)$.

That is, $6|(x^2 -z^2)$, or $x\sim z$.