Quotient Structure on Subset Product

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

Let $$\mathcal{R}$$ be a congruence for $$\circ$$ on $$S$$.

Then:
 * $$\forall X, Y \in S / \mathcal{R}: X \circ_{\mathcal {P}} Y \subseteq X \circ_{\mathcal{R}} Y$$

where:


 * $$S / \mathcal{R}$$ is the quotient of $S$ by $\mathcal{R}$


 * $$\circ_{\mathcal {P}}$$ is the operation induced on $\mathcal {P} \left({S}\right)$ by $\circ$


 * $$\circ_{\mathcal{R}}$$ is the operation induced on $S / \mathcal{R}$ by $\circ$

Proof
By definition of subset product:


 * $$X \circ_{\mathcal P} Y = \left\{{x \circ y: x \in X, y \in Y}\right\}$$

Thus:


 * $$X \circ_\mathcal{R} Y = \left\{{x \circ y: x \in X, y \in Y}\right\} \cup \left\{{x \circ y: x \in \left[\!\left[{X}\right]\!\right]_{\mathcal{R}}, y \in \left[\!\left[{Y}\right]\!\right]_{\mathcal{R}}}\right\}$$.

The result follows from Subset of Union.