Cardinality of Set Union/Corollary

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Theorem

Let $S_1, S_2, \ldots, S_n$ be finite sets which are pairwise disjoint.


Then:

$\displaystyle \card {\bigcup_{i \mathop = 1}^n S_i} = \sum_{i \mathop = 1}^n \card {S_i}$


Specifically:

$\card {S_1 \cup S_2} = \card {S_1} + \card {S_2}$


Proof

As $S_1, S_2, \ldots, S_n$ are pairwise disjoint, their intersections are all empty.

The Cardinality of Set Union holds, but from Cardinality of Empty Set, all the terms apart from the first vanish.

$\blacksquare$


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