Definition:Operation Induced on Quotient Set

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:

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