Proving that $4 \mid m - n$ is an equivalence relation on $\mathbb{Z}$ I have been able to figure out the the distinct equivalence classes. Now I am having difficulties proving the relation IS an equivalence relation. 
$F$ is the relation defined on $\Bbb Z$ as follows: 

For all $(m, n) \in \Bbb Z^2,\ m F n \iff  4 | (m-n)$

equivalence classes:
$\{-8,-4,0,4,8\}, \{-7,-3,1,5,9\} \{-6,-2,2,6,10\}$ and $\{-5, -1, 3, 7, 11\}$

1) reflexivity: $mFm $ since $4|0$
2) simmetry: $mFn \Rightarrow  nFm$ since $4|\pm (m-n)$
3) transitivity: if $4|(m-n)$ and $4|(n-r)$ then $4|\big((m-n)+(n-r)\big)$ or $4|(m-r)$
