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 $$\mathcal R$$:
 * $$S / \mathcal R \ \stackrel {\mathbf {def}} {=\!=} \ \left\{{\left[\!\left[{x}\right]\!\right]_{\mathcal R}: x \in S}\right\}$$

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

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

where $$\mathcal P \left({S}\right)$$ is the power set of $$S$$.

Notation
The notation used to denote a quotient set varies throughout the literature.

uses $$\overline S$$ for $$S / \mathcal R$$.

Also see

 * Quotient Mapping, also known as the Canonical Surjection