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

The quotient mapping is always a surjection, and is often referred to as the canonical surjection or natural surjection from $S$ to $S / \mathcal R$.

Some sources refer to the quotient mapping as the classifying map or classifying mapping, as it classifies the elements of $S$ into those various equivalence classes.

Also see

 * Induced Equivalence

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