Definition:Operation Induced on Quotient Set

Definition
Let $$\mathcal R$$ be a congruence on an algebraic structure $$\left({S, \circ}\right)$$.

The operation $$\circ_{\mathcal R}$$ induced on $$S / \mathcal R$$ (the quotient set of $S$ by $\mathcal R$) by $$\circ$$ is defined as:


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