# Subset Product within Commutative Structure is Commutative

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## Contents

## 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 $\powerset S$.

$\blacksquare$

## Also see

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 9$