Help with equivalence classes for $x\sim$y iff $x-y\in\mathbb{Q}$.

Question

Help with equivalence classes for $x\sim$y iff $x-y\in\mathbb{Q}$.

I need to show the equivalence classes for $[0]_{\sim}$ and $[\sqrt{2}]_{\sim}$.

Here is what I did: $[0]_{\sim}$ = {$a\in\mathbb{X}$|a$\sim$0} = {$a\in\mathbb{X}$|a-0} = {a}.

I haven't tried anything for $\sqrt{2}$ because I could not think of an idea that made sense.

Am on I on the right track? If not how should have I proceeded?

Answer 1

$[0] = \{ x \in \mathbb R \ | \ x - 0 \in \mathbb Q \}$. That is

$$[0] = \{ x \in \mathbb R \ | \ x \in \mathbb Q \} = \mathbb Q$$

Now try again to write down $[\sqrt 2]$.