Given a universal set $S$ and $n$ sets $A_1,A_2,\dots ,A_n$ that are subsets of $S$

Inclusion–exclusion principle

$\left|{\bigcup }_{i=1}^n{A}_i\right|={\sum }_{i=1}^n|A_i|-{\sum }_{1⩽i<j⩽n}|A_i\cap A_j|+{\sum }_{1⩽i<j<k⩽n}|A_i\cap A_j\cap A_k|-\cdots +(-1{)}^{n+1}\left|A_1\cap \cdots \cap A_n\right|$

Intersction form

$\left|{\bigcap }_{i=1}^n{A}_i^{\complement }\right|=\left|S\setminus {\bigcup }_{i=1}^n{A}_i\right|=|S|-{\sum }_{i=1}^n|A_i|+{\sum }_{1⩽i<j⩽n}|A_i\cap A_j|-\cdots +(-1{)}^n|A_1\cap \cdots \cap A_n|$ or $\left|{\bigcap }_{i=1}^n{A}_i\right|=\left|S\setminus {\bigcup }_{i=1}^n{A}_i^{\complement }\right|=|S|-{\sum }_{i=1}^n|A_i^{\complement }|+{\sum }_{1⩽i<j⩽n}|A_i^{\complement }\cap A_j^{\complement }|-\cdots +(-1{)}^n|A_1^{\complement }\cap \cdots \cap A_n^{\complement }|$

The number of elements $A_i$ that are not in any of the other sets

$\left|A_i\setminus \left(\bigcup _{j\ne i}{A}_j\right)\right|=|A_i|-\sum _{j\ne i}|A_i\cap A_j|+\sum _{1⩽j<k⩽n}|A_i\cap A_j\cap A_k|-\cdots$

The number of elements that are in exactly one set

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