Definition:Quotient Set

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Definition

Let $\RR$ be an equivalence relation on a set $S$.

For any $x \in S$, let $\eqclass x \RR$ be the $\RR$-equivalence class of $x$.


The quotient set of $S$ induced by $\RR$ is the set $S / \RR$ of $\RR$-classes of $\RR$:

$S / \RR := \set {\eqclass x \RR: x \in S}$


Also known as

The quotient set of $S$ induced by $\RR$ can also be referred to as:

the quotient of $S$ determined by $\RR$
the quotient of $S$ by $\RR$
the quotient of $S$ modulo $\RR$


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


If $\PP = S / \RR$ is the partition formed by $\RR$, the quotient set can be denoted $S / \PP$.


Also see

  • Results about quotient sets can be found here.


Linguistic Note

The word quotient derives from the Latin word meaning how often.


Sources

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