I need help solving this task, if anyone had a similar problem it would help me.
The task is :
Show that the relation $\rho$ is introduced on the set $\mathbb{Z}$ in the following way $(x,y)\in\rho \iff $exists $k\in\mathbb{Z}$ such that it is $x-y=nk$ where is $n\ge 2$ natural number. Determine the relation of equivalence and equivalence class.
I tried this:
For relation equivalence:
Reflexivity:
$x-x=nk\iff 0=nk \iff k=0, x\rho x$
Symmetry:
$x-y=nk\iff -y+x=nk, y\rho x$
Transitivity:
$x \rho y \land y\rho z\\x-y=nk \land -z+y=nk\\(x-y)+(-z+y)=nk\\x-z=nk$
I don't know if this is good at all? I don't even know how to start equivalence classes.. Thanks in advance !