Proving that rational equivalence is an equivalence relation on any set.

Question

I seek to prove that the rational equivalence relation is an equivalence relation, in that it is reflexive, symmetric, and transitive.

The rational equivalence relation is as follows "Two numbers in a set are rationally equivalent provided their difference is rational".

I know that it is reflexive, since for two points in a set E, a and b, if |a-b| is rational then |b-a| is also rational, because the two are equivalent, and similarly for the irrational case.

I'm not sure how to prove symmetry and transitivity.

Answer 1

Two observations:

1. The absolute value signs are a needless distraction: $|a|$ is rational if and only if $a$ is rational.

2. The transitive property is a consequence of the fact that the sum of two rational numbers is rational.