# Definition:Quotient Set

## Contents

## Definition

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

For any $x \in S$, let $\eqclass x {\mathcal R}$ be the $\mathcal R$-equivalence class of $x$.

Then:

- The
**quotient set of $S$ determined by $\mathcal R$**

or:

- the
**quotient of $S$ by $\mathcal R$**

or:

- the
**quotient of $S$ modulo $\mathcal R$**

is the set $S / \mathcal R$ of $\mathcal R$-classes of $\mathcal R$:

- $S / \mathcal R := \set {\eqclass x {\mathcal R}: x \in S}$

Note that the **quotient set** is a set of sets — each element of $S / \mathcal R$ is itself a set.

In fact:

- $S / \mathcal R \subseteq \powerset S$

where $\powerset S$ is the power set of $S$.

Alternatively, if $\mathcal P = S / \mathcal R$ is the partition formed by $\mathcal R$, the **quotient set** can be denoted $S / \mathcal P$.

## Also denoted as

The notation $\overline S$ can occasionally be seen for $S / \mathcal R$.

## Also see

## Sources

- 1951: Nathan Jacobson:
*Lectures in Abstract Algebra: I. Basic Concepts*... (previous) ... (next): Introduction $\S 3$: Equivalence relations - 1960: Paul R. Halmos:
*Naive Set Theory*... (previous) ... (next): $\S 7$: Relations - 1962: Bert Mendelson:
*Introduction to Topology*... (previous) ... (next): $\S 1.7$: Relations - 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 - 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): $\S 6$ - 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): $\S 1.4$: Equivalence Relations - 2010: Steve Awodey:
*Category Theory*... (previous) ... (next): $\S 3.4$