Finite Union of Finite Sets is Finite/Proof 1

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Theorem

Let $S$ be a finite set of finite sets.


Then the union of $S$ is finite.


Proof

The proof proceeds by induction.

Let $S$ be a finite set with cardinality $n$.

If $n = 0$ then $S = \O$, so $\bigcup S = \O$, which is finite.

Suppose that an arbitrary finite set with cardinality $n$ of finite sets has a finite union.


Let $S$ have cardinality $n^+$.

Then there is a bijection $f: n^+ \to S$.

Then:

$\ds \bigcup S = \bigcup_{k \mathop \in n^+} \map f k = \bigcup_{k \mathop \in n} \map f k \cup \map f n$



By Union of Finite Sets is Finite, $\bigcup S$ is finite.

$\blacksquare$


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