# Subset Product within Commutative Structure is Commutative

## Theorem

Let $\struct {S, \circ}$ be a magma.

If $\circ$ is commutative, then the operation $\circ_\mathcal P$ induced on the power set of $S$ is also commutative.

## Proof

Let $\struct {S, \circ}$ be a magma in which $\circ$ is commutative.

Let $X, Y \in \powerset S$.

Then:

 $\displaystyle X \circ_\mathcal P Y$ $=$ $\displaystyle \set {x \circ y: x \in X, y \in Y}$ $\displaystyle Y \circ_\mathcal P X$ $=$ $\displaystyle \set {y \circ x: x \in X, y \in Y}$

from which it follows that $\circ_\mathcal P$ is commutative on $\powerset S$.

$\blacksquare$