# Definition:Quotient Mapping

## Contents

## 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**.

## Examples

### Congruence Modulo 3

Let $x \mathrel \RR y$ be the equivalence relation defined on the integers as congruence modulo $3$:

- $x \mathrel \RR y \iff x \equiv y \pmod 3$

defined as:

- $\forall x, y \in \Z: x \equiv y \pmod 3 \iff \exists k \in \Z: x - y = 3 k$

That is, if their difference $x - y$ is a multiple of $3$.

From Congruence Modulo $3$, the quotient set induced by $\RR$ is:

- $\Z / \RR = \set {\eqclass 0 3, \eqclass 1 3, \eqclass 2 3}$

Hence the **quotient mapping** $q_\RR: \Z \to \Z / \RR$ is defined as:

- $\forall x \in \Z: \map {q_\RR} x = \eqclass x 3 = \set {x + 3 k: k \in \Z}$

### Modulo 2 pi as Angular Measurement

Let $\RR$ denote the congruence relation modulo $2 \pi$ on the real numbers $\R$ defined as:

- $\forall x, y \in \R: \tuple {x, y} \in \RR \iff \text {$x$ and $y$}$ measure the same angle in radians

From Congruence Modulo $2 \pi$ as Angular Measurement, the quotient set induced by $\RR$ is:

- $\R / \RR = \set {\eqclass \theta {2 \pi}: 0 \le \theta < 2 \pi}$

where:

- $\eqclass \theta {2 \pi} = \set {\theta + 2 k \pi: k \in \Z}$

Hence the **quotient mapping** $q_\RR: \R \to \R / \RR$ is defined as:

- $\forall x \in \R: \map {q_\RR} x = \eqclass x {2 \pi} = \set {x + 2 k \pi: k \in \Z}$

## Also see

- Results about
**quotient mappings**can be found here.

## Sources

- 1951: Nathan Jacobson:
*Lectures in Abstract Algebra: Volume $\text { I }$: Basic Concepts*... (previous) ... (next): Introduction $\S 3$: Equivalence relations - 1960: Paul R. Halmos:
*Naive Set Theory*... (previous) ... (next): $\S 8$: Functions - 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 10$ - 1967: George McCarty:
*Topology: An Introduction with Application to Topological Groups*... (previous) ... (next): Chapter $\text{I}$: Sets and Functions: Quotient Functions - 1968: A.N. Kolmogorov and S.V. Fomin:
*Introductory Real Analysis*... (previous) ... (next): $\S 1.4$: Decomposition of a set into classes. Equivalence relations: Example $2$ - 1971: Allan Clark:
*Elements of Abstract Algebra*... (previous) ... (next): Chapter $1$: Equivalence Relations: $\S 17$ - 1972: A.G. Howson:
*A Handbook of Terms used in Algebra and Analysis*... (previous) ... (next): $\S 3$: Equivalence relations and quotient sets: Quotient sets - 1975: T.S. Blyth:
*Set Theory and Abstract Algebra*... (previous) ... (next): $\S 6$. Indexed families; partitions; equivalence relations - 1975: Bert Mendelson:
*Introduction to Topology*(3rd ed.) ... (previous) ... (next): Chapter $1$: Theory of Sets: $\S 7$: Relations - 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.4$: Equivalence relations - 2010: Steve Awodey:
*Category Theory*(2nd ed.) ... (previous) ... (next): $\S 3.4$