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:

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

Also see

 * Subset Product within Semigroup is Associative