Let $\sim$ be a relation on $\mathbb{R}$ and $x\sim y$ if and only if $x-y\in \mathbb{Z}$.
(a) Show that $\sim$ is an equivalence relation
(b) Give a complete set of equivalence class representatives.
(a) is easy to show, but I really don’t understand (b).
I know that:
$[a]_{\sim}:=\{y\in \mathbb{R} \mid a\sim y\}$
so
$[0]_{\sim}=\{\dots,-2,-1,0,1,2,\dots\}$
and that
$\dots=[-2]_{\sim}=[-1]_{\sim}=[0]_{\sim}=[1]_{\sim}=[2]_{\sim}=\,\dots$
because if $x\sim y$ then $[x]=[y]$.
But how can I find a complete set of equivalence class representatives?
Thanks in advance!