Definition:Quotient Mapping

Definition
Let $$\mathcal{R} \subseteq S \times S$$ be an equivalence on a set $$S$$.

Let $$\left[\!\left[{s}\right]\!\right]_{\mathcal{R}}$$ be the $\mathcal{R}$-equivalence class of $$s$$.

Let $$S / \mathcal{R}$$ be the quotient set of $$S$$ determined by $$\mathcal{R}$$.

Then $$q_{\mathcal{R}}: S \to S / \mathcal{R}$$ is the quotient mapping induced by $$\mathcal{R}$$, and is defined as:


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

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 $$q \left({s}\right)$$.

This mapping is always a surjection, and is often referred to as the Canonical Surjection from $$S$$ to $$S / \mathcal{R}$$.