Congruences on Rational Numbers

Theorem
There are only two congruence relations on the field of rational numbers $$\left({\Q, +, \times}\right)$$:


 * 1) The diagonal relation $$\Delta_{\Q}$$;
 * 2) The trivial relation $$\Q\times \Q$$.

Proof
From: we know that both these relations are compatible with both addition and multiplication on $$\Q$$.
 * Diagonal Relation is Universally Compatible and
 * Trivial Relation is Universally Congruent

Now we need to show that these are the only such relations.

Let $$\mathcal{R}$$ be a congruence on $$\Q$$, such that $$\mathcal{R} \ne \Delta_{\Q}$$.

$$ $$ $$ $$

Then:

$$ $$ $$ $$ $$