# Definition:Quotient Mapping

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## 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**.

## Also see

## 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): $\text{I}$: 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): $\S 3.4$