# Definition:Quotient Set

## Definition

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

For any $x \in S$, let $\eqclass x \RR$ be the $\RR$-equivalence class of $x$.

The quotient set of $S$ induced by $\RR$ is the set $S / \RR$ of $\RR$-classes of $\RR$:

$S / \RR := \set {\eqclass x \RR: x \in S}$

## Also known as

The quotient set of $S$ induced by $\RR$ can also be referred to as:

the quotient of $S$ determined by $\RR$
the quotient of $S$ by $\RR$
the quotient of $S$ modulo $\RR$

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

If $\PP = S / \RR$ is the partition formed by $\RR$, the quotient set can be denoted $S / \PP$.