Definition:Quotient Set

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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


Sources