Finding the equivalence classes if $\left\{ (x,y)\,:\,x\equiv y\mod5\right\}$

Question

I have the following relation $R\subseteq\mathbb{N}\times\mathbb{N}$: $$R=\left\{ (x,y)\,:\,x\equiv y\mod5\right\} $$ I have proved that $R$ is a equivalence relation. I would like to find the equivalence classes. As I understand the classes are $[i]$ where $i\in \{0,\ldots,4\}$. I'm struggling of proving it formally. Is it possible to show how to prove it formally?

Answer 1

You have to show that these classes are disjoint and their union is the set of natural numbers. Make sure to chose $5$ instead of $0$ as the representative of that class.

Answer 2

Yes, it is correct. If $x\in\mathbb N$, then let $r_x$ be the remainder of the division of $x$ by $5$. Then:

1. $r_x\in\{0,1,2,3,4\}$;
2. $x\equiv r_x$.

Furthermore, if $i,j$ are distinct elements of $\{0,1,2,3,4\}$, then $i\not\equiv j$.

Answer 3

It can be proved that an equivalence relation partitions the set it's defined on.

In this case, the equivalence classes partition $\Bbb N$ into $5$ equivalence classes. For each $n\in\Bbb N$, $\exists! k\in\{0,1,2,3,4\}$ such that $n=k+5l$ for some $l$.

This is the result of Euclidean division.