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 Equivalence Class is Not Empty, 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.