# Definition:Quotient Mapping

## Definition

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

Let $\eqclass s {\mathcal R}$ 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:

$q_\mathcal R: S \to S / \mathcal R: \map {q_\mathcal R} s = \eqclass s {\mathcal R}$

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

If the equivalence $\mathcal R$ is understood, $\map {q_\mathcal R} s$ can be written $\map q s$.

## Also known as

The quotient mapping is often referred to as:

the canonical surjection from $S$ to $S / \mathcal R$
the canonical map or canonical projection from $S$ onto $S / \mathcal R$
the natural mapping 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)
the projection from $S$ to $S / \mathcal R$

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

Some sources use $\pi$ to denote the quotient mapping.