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 $$x$$.

The quotient set of $$S$$ determined by $$\mathcal{R}$$, or the quotient of $$S$$ by $$\mathcal{R}$$ 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 $$S / \mathcal{R}$$.