Subset Product within Commutative Structure is Commutative

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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 $\powerset S$.

$\blacksquare$


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