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 or natural surjection from $$S$$ to $$S / \mathcal R$$.

Notation
Some sources denote the quotient mapping by $$\natural_{\mathcal R}$$. This is logical, as $$\natural$$ is the "natural" sign in music.