Quotient Structure on Subset Product

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

Let $\RR$ be a congruence for $\circ$ on $S$.

Then:
 * $\forall X, Y \in S / \RR: X \circ_\PP Y \subseteq X \circ_\RR Y$

where:


 * $S / \RR$ is the quotient of $S$ by $\RR$


 * $\circ_\PP$ is the operation induced on $\powerset S$ by $\circ$


 * $\circ_\RR$ is the operation induced on $S / \RR$ by $\circ$

Proof
By definition of subset product:


 * $X \circ_\PP Y = \set {x \circ y: x \in X, y \in Y}$

Thus:


 * $X \circ_\RR Y = \set {x \circ y: x \in X, y \in Y} \cup \set {x \circ y: x \in \eqclass X \RR, y \in \eqclass Y \RR}$

The result follows from Subset of Union.