Subset Product within Commutative Structure is Commutative

Theorem
Let $\left({S, \circ}\right)$ 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 $\left({S, \circ}\right)$ be a magma in which $\circ$ is commutative.

Let $X, Y \in \mathcal P \left({S}\right)$.

Then:


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


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

from which it follows that $\circ_\mathcal P$ is commutative on $\mathcal P \left({S}\right)$.

Also see

 * Subset Product within Semigroup is Associative