Definition:Operation Induced on Quotient Set

Definition
Let $\left({S, \circ}\right)$ be an algebraic structure.

Let $\mathcal R$ be a congruence relation on $\left({S, \circ}\right)$.

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:


 * $\left[\!\left[{x}\right]\!\right]_\mathcal R \circ_\mathcal R \left[\!\left[{y}\right]\!\right]_\mathcal R = \left[\!\left[{x \circ y}\right]\!\right]_\mathcal R$

Also see

 * Quotient Structure
 * Quotient Structure is Well-Defined