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:
- $\ds \card {\bigcup_{i \mathop = 1}^n S_i} = \sum_{i \mathop = 1}^n \card {S_i}$
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.
$\blacksquare$
Sources
- 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 13$: Arithmetic
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $1$: Sets and Logic: Exercise $8 \ \text{(a)}$
- 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.2$: Sets: Exercise $3$