The following proposition is a generalization of the previous lemma (no. 1) .

Proposition: Counting the Set's Elements Using Its Partition

Let \(S\) be a finite set and let its subsets \(S_1,\ldots S_n\subset S\) form a partition of \(S\), i.e.

\[S=\bigcup_{i=1}^n S_i,~~~~~~S_i\cap S_j=\emptyset,\forall i\neq j.\]

Further let the cardinalities \(|S_i|\) be known for all \(i=1,\ldots n\). Then the cardinality of the set \(S\) is given by the sum of the known cardinalities of its subsets.

\[|S|=\sum_{i=1}^n |S_i|.\]

Proofs: 1

Thank you to the contributors under CC BY-SA 4.0!




  1. Aigner, Martin: "Diskrete Mathematik", vieweg studium, 1993