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:

$$ $$ $$ $$ $$