# Definition:Quotient Mapping

## Definition

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

Let $\eqclass s \RR$ be the $\RR$-equivalence class of $s$.

Let $S / \RR$ be the quotient set of $S$ determined by $\RR$.

Then $q_\RR: S \to S / \RR$ is the quotient mapping induced by $\RR$, and is defined as:

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

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

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

## Also known as

The quotient mapping is often referred to as:

the canonical surjection from $S$ to $S / \RR$
the canonical map or canonical projection from $S$ onto $S / \RR$
the natural mapping from $S$ to $S / \RR$
the natural surjection from $S$ to $S / \RR$
the classifying map or classifying mapping (as it classifies the elements of $S$ into those various equivalence classes)
the projection from $S$ to $S / \RR$

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

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