Difference is Rational is Equivalence Relation

Theorem
Define $\sim$ as the relation on real numbers given by:


 * $x \sim y \iff x - y \in \Q$

That is, that the difference between $x$ and $y$ is rational.

Then $\sim$ is an equivalence relation.

Proof
Checking in turn each of the criteria for equivalence:

Reflexive
So $\sim$ is reflexive.

Symmetric
So $\sim$ is symmetric.

Transitive
So $\sim$ is transitive.

Thus $\sim$ has been shown to be reflexive, symmetric and transitive.

Hence by definition it is an equivalence relation.