Power Structure of Subset is Closed iff Subset is Closed
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Theorem
Let $\struct {S, \circ}$ be a magma.
Let $\circ_\PP$ be the operation induced on $\powerset S$, the power set of $S$.
Let $T \subseteq S$.
Then:
- the algebraic structure $\struct {\powerset T, \circ_\PP}$ is closed
- the algebraic structure $\struct {T, \circ}$ is closed.
Proof
Sufficient Condition
Let $\struct {\powerset T, \circ_\PP}$ be closed.
Then:
\(\ds \forall X, Y \in \powerset T: \, \) | \(\ds X \circ_\PP Y\) | \(\in\) | \(\ds \powerset T\) | Definition of Closed Algebraic Structure | ||||||||||
\(\ds \set {x \circ y: x \in X, y \in Y}\) | \(\subseteq\) | \(\ds T\) | Definition of Operation Induced on Power Set | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \forall x \in X, y \in Y: \, \) | \(\ds x \circ y\) | \(\in\) | \(\ds T\) | Definition of Subset |
That is, $\struct {T, \circ}$ is closed.
$\Box$
Necessary Condition
Let $\struct {T, \circ}$ be closed.
Then by definition $\struct {T, \circ}$ is a magma.
From Power Structure of Magma is Magma it follows that $\struct {\powerset T, \circ_\PP}$ is likewise a magma.
That is, the algebraic structure $\struct {\powerset T, \circ_\PP}$ is closed.
$\blacksquare$
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: Exercise $9.8$