# Describe its equivalence classes of $x-y\in\mathbb{Z}$

For $$x,y\in\mathbb{R}$$ define x ~ y to mean that $$x-y\in\mathbb{Z}$$. Prove that ~ is an equivalence relation on $$\mathbb{R}$$. Describe its equivalence classes.

I've successfully proved x ~ y relation is reflexive, symmetric and transitive.

What I'm not able to do is to describe the equivalence classes. In my view it can be

• $$\mathbb{Z}$$ as every integer $$x,y$$ is $$x~R~y$$ as demonstrated above

• or more specifically even $$\mathbb{R}$$ given that $$x-y\in\mathbb{Z}$$.

Am I missing something?

• Given a single $x\in \Bbb R$, its equivalence class under $\sim$ is given by those $y$ such that $y = x+n$ where $n\in \Bbb Z$. How would you write this in set-builder notation? Can you pick a nice interval to work in for a simple description..? Commented Dec 20, 2019 at 15:30
• Try some examples to get a feel for it. For instance, if $y = 1.27$ then $x \sim 1.27 \iff x - 1.27 \in \mathbb Z \iff \exists n \in \mathbb Z$ such that $x-1.27 = n \iff \exists n \in \mathbb Z$ such that $x = 1.27+n$. Now substitute some values of $n$ into $x=1.27+n$, like $n=0,1,2,3$ and $n=-1,-2,-3$. See if you observe a pattern, and see if you can generalize that pattern. Commented Dec 20, 2019 at 15:43
• I think I can see it now. It's pretty much what has been given by in the answer section, so I guess it's $\{x+n\in\mathbb{Z}|x\in[0,1)\}$. So it's not complete $\mathbb{R}$, but something close enough.
– user620319
Commented Dec 20, 2019 at 15:48

Each equivalence class is a subset of $$\mathbb R$$ and therefore the set of all equivalence classes must be a set of subsets of $$\mathbb R$$.
In your case, for each $$x\in[0,1)$$, consider the set $$x+\mathbb Z$$. There you have it. The set of all equivalence classes is$$\{x+\mathbb Z\mid x\in[0,1)\}.\tag1$$Note that if $$x\in\mathbb R$$, then $$x-\lfloor x\rfloor\in[0,1)$$, that $$x\sim x-\lfloor x\rfloor$$ and that $$x-\lfloor x\rfloor$$ is the only $$y\in[0,1)$$ such that $$x\sim y$$. Therefore, $$(1)$$ describes all equivalence classes and each equivalence class appears there only once.