Complete set of equivalence class representative 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!

Hint: two numbers are equivalent under this relation iff they have the same fractional part.
