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

if and only if:

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