Cardinality of Set Union/3 Sets

Theorem

Let $S_1$, $S_2$ and $S_3$ be finite sets.

Then:

\(\ds \card {S_1 \cup S_2 \cup S_3}\)

\(=\)

\(\ds \card {S_1} + \card {S_2} + \card {S_3}\)

\(\ds \)

\(\)

\(\, \ds - \, \)

\(\ds \card {S_1 \cap S_2} - \card {S_1 \cap S_3} - \card {S_2 \cap S_3}\)

\(\ds \)

\(\)

\(\, \ds + \, \)

\(\ds \card {S_1 \cap S_2 \cap S_3}\)

Proof

This is a specific example of Cardinality of Set Union: General Case.

$\blacksquare$

Sources

• 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 1$. Sets; inclusion; intersection; union; complementation; number systems: Exercise $8$
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): cardinality