Principle of Inclusion-Exclusion

The Principle of Inclusion-Exclusion (abbreviated PIE) provides an organized method/formula to find the number of elements in the union of a given group of sets, the size of each set, and the size of all possible intersections among the sets.

Remarks

Sometimes it is also useful to know that, if you take into account only the first $m\le n$ sums on the right, then you will get an overestimate if $m$ is odd and an underestimate if $m$ is even. So,

$\left|\bigcup_{i=1}^n A_i\right|\le \sum_{i=1}^n\left|A_i\right|$ ,

$\left|\bigcup_{i=1}^n A_i\right|\ge \sum_{i=1}^n\left|A_i\right|-\sum_{i < j}\left|A_i\cap A_j\right|$ ,

$\left|\bigcup_{i=1}^n A_i\right|\le \sum_{i=1}^n\left|A_i\right|-\sum_{i < j}\left|A_i\cap A_j\right| +\sum_{i<j<k}\left|A_i\cap A_j\cap A_k\right|$ ,

and so on.

Examples

2002 AIME I Problems/Problem 1 http://artofproblemsolving.com/wiki/index.php?title=2002_AIME_I_Problems/Problem_1#Problem

2011 AMC 8 Problems/Problem 6 https://artofproblemsolving.com/wiki/index.php?title=2011_AMC_8_Problems/Problem_6

2017 AMC 10B Problems/Problem 13 https://artofproblemsolving.com/wiki/index.php?title=2017_AMC_10B_Problems/Problem_13

2005 AMC 12A Problems/Problem 18 https://artofproblemsolving.com/wiki/index.php/2005_AMC_12A_Problems/Problem_18

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Principle of Inclusion-Exclusion

Remarks

Examples

See also