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 $\circ_\PP$ 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:
\(\ds X \circ_\PP Y\) | \(=\) | \(\ds \set {x \circ y: x \in X, y \in Y}\) | Definition of $\circ_\PP$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \set {y \circ x: x \in X, y \in Y}\) | Definition of Commutative Operation | |||||||||||
\(\ds \) | \(=\) | \(\ds Y \circ_\PP X\) | Definition of $\circ_\PP$ |
Hence $\circ_\PP$ is commutative on $\powerset S$.
$\blacksquare$
Also see
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 9$: Compositions Induced on the Set of All Subsets