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.