Equivalence relations-Discrete Math

Question

Here is an equivalence relation R={ (x,y) | x-y is an integer}

My question is: what is the equivalence class of 1 for this equivalence relation?

Can say indicate the equivalence class of 1 as [(1)] = { (x,y), x-y= }

I am confused about how to write the right hand side? can someone help me?

Answer 1

The equivalence class of $1$ is $\mathbb{Z}$. Show that $x-1$ is an integer if and only if $x$ is an integer.

Answer 2

It seems like your equivalence relation is that $x$ is related to $y$ if $x-y \in \mathbb Z$. What numbers give integers if you subtract $1$ from them? The integers, since they are closed under subtraction.