Attempting to find the equivalence class of 5.

Question

For $a,b \in \mathbb{R}$ define $a \sim b$ if $a - b \in \mathbb{Z}$

How would you find the equivalence class of 5. In other words what I'm trying to describe is the set $[5]$ = {$y : 5 \sim y$}. And $[5]$ is just the name of the set.

Answer 1

HINT: By definition $[5]=\{x\in\Bbb R:5\sim x\}=\{x\in\Bbb R:5-x\in\Bbb Z\}$. $5-x\in\Bbb Z$ if and only if $5-x$ is an integer, i.e., there is an $n\in\Bbb Z$ such that $5-x=n$. By elementary algebra this is equivalent to saying that $x=5-n$ for some integer $n$, so $[5]=\{5-n:n\in\Bbb Z\}$. What real numbers can be written in the form $5-n$ for some integer $n$? This set has a very simple description.

Answer 2

The answers to your previous question contain a description of the equivalence classes:

$a \sim b$ iff $a$ and $b$ have the same fractional part.

The fractional part of $5$ is $0$. Which other real numbers have zero fractional part?