Congruences on Rational Numbers

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


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

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

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

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