Definition:Quotient Set

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

For any $x \in S$, let $\left[\!\left[{x}\right]\!\right]_\mathcal R$ 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 $\mathcal R$:
 * $S / \mathcal R := \left\{{\left[\!\left[{x}\right]\!\right]_\mathcal R: x \in S}\right\}$

Note that the quotient set is a set of sets &mdash; each element of $S / \mathcal R$ is itself a set.

In fact:
 * $S / \mathcal R \subseteq \mathcal P \left({S}\right)$

where $\mathcal P \left({S}\right)$ 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

 * Fundamental Theorem on Equivalence Relations


 * Definition:Quotient Mapping
 * Definition:Quotient Structure
 * Definition:Quotient Group