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

Alternative names
The quotient mapping is often referred to as:
 * the canonical surjection from $S$ to $S / \mathcal R$
 * the natural surjection from $S$ to $S / \mathcal R$
 * the classifying map or classifying mapping (as it classifies the elements of $S$ into those various equivalence classes)

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

Also see

 * Quotient Mapping is Surjection
 * Induced Equivalence