Equivalence class of a number on a relation?

Question

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?

Answer 1

Your $x$ is fixed and equal to $1/3$.

Let $x-y=k\in Z$ , then $y=x-k=\frac{1}{3}-k$ is the equivalence class you're looking for. $1/3$ is just one element of it.

Answer 2

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 $n\in\mathbb{Z}$.