# Definition:Quotient Set

## 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 $S / \mathcal R$.