Equivalence class of a number on a relation? 
Lets say there is an equivalence relation $x\sim y$ if and only if $x-y$ is an integer. Find the equivalence class of the number $\frac13$.

I came up with $\left[\frac13\right]=\left\{\frac13\right\}$ but I'm not sure if its right. Any tips?

Hint: Let $S=\{(x,y)\mid x-y\in \mathbb{Z}\}$. Suppose that $(1/3,y)\in S$, then $1/3-y=n$ for some $n\in\mathbb{Z}$ and $y=1/3-n$. Moreover, if $y=1/3-n$ for some $n\in\mathbb{Z}$ then $(1/3,y)\in \mathbb{Z}$. Thus $(1/3,y)\in S$ if and only if $y=1/3-n$ for some $n\in\mathbb{Z}$.
