Definition:Quotient Mapping

For any equivalence $$\mathcal{R} \subseteq S \times S$$ on a set $$S$$, the quotient mapping $$q_{\mathcal{R}}: S \to S / \mathcal{R}$$ is the mapping from $$S$$ to $$S / \mathcal{R}$$ defined as:

$$q_{\mathcal{R}}: S \to S / \mathcal{R}: q_{\mathcal{R}} \left({s}\right) = \left[\left[{s}\right]\right]_{\mathcal{R}}$$

where $$\left[\left[{s}\right]\right]_{\mathcal{R}}$$ is the $\mathcal{R}$-equivalence class of $$s$$, and $$S / \mathcal{R}$$ is the quotient set of $$S$$ determined by $$\mathcal{R}$$.

Effectively, we are defining a mapping on $$S$$ by assigning each element $$s \in S$$ to its equivalence class $$\left[\left[{s}\right]\right]_{\mathcal{R}}$$.

If the equivalence $$\mathcal{R}$$ is understood, $$q_{\mathcal{R}} \left({s}\right)$$ can be written $$q \left({s}\right)$$.