Equivalence relations and their class

Question

I am trying to figure out how to prove that an equivalence relation of the relation x~y defined on Z <=> x-y is a multiple of 3.

My attempt was:

1) Reflexive: x = x => x ~ x

2) Symmetric: x ~ y => x - y => - y + x => y ~ x

3) Transitive: ... <--- how do I make use of the fact that x-y is a multiple of 3?

Thanks for help!

Answer 1

You have not proven anything.

$x-x=0$ is a multiple of $3$, so it is reflexive.

If $x-y=3k$ for some integer $k$, then $y-x=-3k$ is a multiple of $3$, so it is symmetric.

If $x-y=3m$ for some integer $m$ and $y-z=3n$ for some integer $n$, then $x-z=3(m+n)$ is a multiple of $3$, so it is transitive.