Editing Combinatorial principles (section)

== Inclusion–exclusion principle ==

[[File:Inclusion-exclusion.svg|thumb|Inclusion–exclusion illustrated for three sets]]

{{main|Inclusion–exclusion principle}}

The inclusion–exclusion principle relates the size of the union of multiple sets, the size of each set, and the size of each possible intersection of the sets. The smallest example is when there are two sets: the number of elements in the union of ''A'' and ''B'' is equal to the sum of the number of elements in ''A'' and ''B'', minus the number of elements in their intersection.

Generally, according to this principle, if ''A''<sub>1</sub>, …, ''A<sub>n</sub>'' are finite sets, then

:<math>\begin{align}
\left|\bigcup_{i=1}^n A_i\right|
& {} = \sum_{i=1}^n \left|A_i\right|
-\sum_{i,j\,:\,1 \le i < j \le n} \left|A_i\cap A_j\right| \\
& {}\qquad +\sum_{i,j,k\,:\,1 \le i < j < k \le n} \left|A_i\cap A_j\cap A_k\right|-\ \cdots\ + \left(-1\right)^{n-1} \left|A_1\cap\cdots\cap A_n\right|.
\end{align}</math>