Define a relation $ \sim $ on the set of real numbers as follows: For $x, y \in \mathbb{R}$ : $x \sim y$ if $ −∈\mathbb{Q}$
I have proven that this is an equivalence class:
Reflexitivity
$x - x = 0$
$0 \in \mathbb{Q}$
so $x\sim x$
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 class of $x$ and most importantly, how to construct an explicit bijection from the equivalence class, $[x]$, to $\mathbb{Q}$.