Define a relation $ sim $ on the set of real numbers as follows: For $x, y in mathbb{R}$ : $x sim y$ if $𝑥 −𝑦∈mathbb{Q}$

I have proven that this is an equivalence class:

**Reflexitivity**

$x – x = 0$

$0 in mathbb{Q}$

so $xsim x$

**Symmetry**

$x – y = a$, where $a in mathbb{Q}$

$y – x = -a$, where $-a in mathbb{Q}$

so $x sim y, y sim x$

**Transitive**

$x – y = a$, $a in mathbb{Q}$,

$y – z = b$, $z in mathbb{R}$, $b in mathbb{Q}$

$x – y + y – z = a + b$

$= x – z = a + b$

$= x – z = c$, $c in mathbb{Q}$

Thus, $x sim y$ is an equivalence relation. However, I am having trouble finding the equivalence classes and most importantly, how to construct an explicit bijection from the equivalence class, $[x]$, to $mathbb{Q}$.