equivalence relation proof {(x, y) ∈ R^2 ∶ x − y ∈ Q}

Question

Prove that {(x, y) ∈ R^2 ∶ x − y ∈ Q} is an equivalence relation on the set of real numbers, where Q denotes the set of rational numbers.

Hi guys, not really sure how to start this question. My first thoughts are to check if its reflexive,symmetric, antisymmetric,transitive. After that, not too sure where to go.

Answer 1

Reflexive

$xRx, x-x= 0$ and $0\in \mathbb Q$

Symmetric

$xRy\implies yRx$ if $x-y$ is rational then $y-x = -(x-y)$ is also rational.

Transitive

$xRy, yRz\implies xRz$

$(x-z) = (x-y) + (y-z)$ and a $\mathbb Q$ is closed under addition so (x-z) must be rational