Definition:Quotient Set

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 $$\left\{{\left[\left[{x}\right]\right]_{\mathcal{R}}: x \in S}\right\}$$ of $$\mathcal{R}$$.

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