Definition:Quotient Set

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

For any $x \in S$, let $\eqclass x {\mathcal R}$ be the $\mathcal R$-equivalence class of $x$.

The quotient set of $S$ induced by $\mathcal R$ is the set $S / \mathcal R$ of $\mathcal R$-classes of $\mathcal R$:
 * $S / \mathcal R := \set {\eqclass x {\mathcal R}: x \in S}$

Also known as
The quotient set of $S$ induced by $\mathcal R$ can also be referred to as:


 * the quotient of $S$ determined by $\mathcal R$
 * the quotient of $S$ by $\mathcal R$
 * the quotient of $S$ modulo $\mathcal R$

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

If $\mathcal P = S / \mathcal R$ is the partition formed by $\mathcal R$, the quotient set can be denoted $S / \mathcal P$.

Also see

 * Fundamental Theorem on Equivalence Relations


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