Cardinality of Set Union/2 Sets

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Theorem

Let $S_1$ and $S_2$ be finite sets.

Then:

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


Proof

We have that Cardinality is Additive Function.

From Additive Function is Strongly Additive:

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

from which the result follows.

$\blacksquare$


Sources

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