# Definition:Operation Induced on Quotient Set

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## Definition

Let $\struct {S, \circ}$ be an algebraic structure.

Let $\mathcal R$ be a congruence relation on $\struct {S, \circ}$.

Let $S / \mathcal R$ be the quotient set of $S$ by $\mathcal R$.

The operation $\circ_\mathcal R$ induced on $S / \mathcal R$ by $\circ$ is defined as:

$\eqclass x {\mathcal R} \circ_\mathcal R \eqclass y {\mathcal R} = \eqclass {x \circ y} {\mathcal R}$