Describe the partiton for the equivalence relation.
For each $x,y\in \mathbb{R}$ xRy $\iff$ $x-y\in \mathbb{Z}$
Now I am not sure how to find a partition for this I guess one could have negative integers or positive integers
I know a the set in a partition family must not overlap, and every element in the original set must be in the partition, and all the element in the partition must be in the original set.
Given a number $x$, we have $yRx \iff y=x+n$ for some integer $n$. So the equivalence class of $x$ is $x+\mathbb{Z}$, that is, {$...,x-2,x-1,x,x+1,x+2,...$}.
The set of all such equivalence classes is the partition formed on $\mathbb{R}$. Now, $xRy$ iff $x$ and $y$ have the same decimal part. For example, $1.23R5.23$. So the distinct equivalence classes correspond precisely to the elements of $[0,1)$. That is, each distinct equivalence class contains exactly one element of this interval.
In short, this relation partitions $\mathbb{R}$ into sets of numbers having the same decimal part.