Q:

Is R = {(a, b) | a, b ∈ Z, ∃k ∈ Z s.t. (a − b) = 3k} an equivalence relation?

I'm not quite sure what ∃k ∈ Z s.t. (a − b) = 3k describes in the relation. Perhaps if I could figure out what that means, I could prove R is reflexive, symmetric, and transitive.
The question also states that if R is an equivalence relation, I should describe its equivalence classes. Could someone explain more clearly to me what equivalence classes are and how to "describe" them?
Thank you in advance.

A:

Hints: 
$(8,5)\in R$ because $8-5=3=3\times 1~~~~$  (and $1\in \mathbb{Z}$).  $(8,2)\in R$ because $8-2=6=3\times 2$.  $(2,8)\in R$ because $2-8=-6=3\times -2$. $(2,2)\in R$ because $2-2=0=3\times 0$.
  However $(8,3)\notin R$ because $8-3=5=3\times (5/3)$, and $5/3\notin \mathbb{Z}$.
Of the three properties you need to prove, the trickiest one is transitivity.  If $(a,b)\in R$ and $(b,c)\in R$, then $a-b=3m$ and $b-c=3n$.  Sum these to get $a-c=3m+3n=3(m+n)$.

