Cardinality of Set Union/Corollary

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 $3$
This article is issued from ProofWiki. The text is available under Creative Commons Attribution-ShareAlike License unless otherwise noted. Additional terms may apply for the media files.