Cardinality of Set Union/General Case

Theorem

Let $\sequence {S_n}_{n \mathop \in \N}$ be a sequence of sets.

Then:

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

$=$

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

$\ds $

$\, \ds - \, $

$\ds \sum_{1 \mathop \le i \mathop < j \mathop \le n} \card {S_i \cap S_j}$

$\ds $

$\, \ds + \, $

$\ds \sum_{1 \mathop \le i \mathop < j \mathop < k \mathop \le n} \card {S_i \cap S_j \cap S_k}$

$\ds $

$\ds \cdots$

$\ds $

$\, \ds + \, $

$\ds \paren {-1}^{n - 1} \card {\bigcap_{i \mathop = 1}^n S_i}$

Proof

By Cardinality is Additive Function, we can apply the Inclusion-Exclusion Principle:

If $f: \SS \to \R$ is an additive function, then:

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$\ds \map f {\bigcup_{i \mathop = 1}^n S_i}$

$=$

$\ds \sum_{i \mathop = 1}^n \map f {S_i}$

$\ds $

$\, \ds - \, $

$\ds \sum_{1 \mathop \le i \mathop < j \mathop \le n} \map f {S_i \cap S_j}$

$\ds $

$\, \ds + \, $

$\ds \sum_{1 \mathop \le i \mathop < j \mathop < k \mathop \le n} \map f {S_i \cap S_j \cap S_k}$

$\ds $

$\ds \cdots$

$\ds $

$\, \ds + \, $

$\ds \paren {-1}^{n - 1} \map f {\bigcap_{i \mathop = 1}^n S_i}$

$\blacksquare$

Sources

1965: J.A. Green: Sets and Groups ... (previous) ... (next): Chapter $1$. Sets: Exercise $4$