Set of Finite Subsets under Induced Operation is Closed

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Theorem

Let $\struct {S, \circ}$ be a magma.

Let $\struct {\powerset S, \circ_\PP}$ be the power structure of $\struct {S, \circ}$

Let $T \subseteq \powerset S$ be the set of all finite subsets of $S$.


Then the algebraic structure $\struct {T, \circ_\PP}$ is closed.


Proof

Let $X, Y \in T$.

Then:

\(\ds X \circ_\PP Y\) \(=\) \(\ds \set {x \circ y: x \in X, y \in Y}\) Definition of Operation Induced on Power Set
\(\ds \leadsto \ \ \) \(\ds \card {X \circ_\PP Y}\) \(\le\) \(\ds \card X \times \card Y\)
\(\ds \leadsto \ \ \) \(\ds X \circ_\PP Y\) \(\in\) \(\ds T\)


This article contains statements that are justified by handwavery.
In particular: Between step $1$ and step $2$ we really need a result Order of Subset Product is not Greater than Cardinality of Cartesian Product which is straightforward but tedious, and I lack patience.
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$\blacksquare$


Sources

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