Quotient Mapping is Surjection

Theorem
Let $$\mathcal R$$ be an equivalence relation on $$S$$.

Then the quotient mapping $$q_{\mathcal R}: S \to S / \mathcal R$$ is a surjection.

It is often referred to as the canonical surjection or the natural surjection from $$S$$ to $$S / \mathcal R$$.

Proof
From No Equivalence Class is Null, we have that:


 * $$\forall \left[\!\left[{x}\right]\!\right]_{\mathcal R} \in S / \mathcal R: \exists x \in S: x \in \left[\!\left[{x}\right]\!\right]_{\mathcal R}$$

... and the result follows.