The inclusion- exclusion principle

The Inclusion-Exclusion Principle

The Inclusion-Exclusion Principle is a fundamental concept in set theory and combinatorics.
This principle is used to calculate the number of elements in unions of several sets.
It caters for the fact that when you add together the sizes of two sets, you count the elements in their intersection twice.

If considering two sets, A and B, the principle is expressed as: A ∪ B = A + B - A ∩ B .
This translates as: the number of elements in A union B equals the number of elements in A plus the number of elements in B minus the number of elements in A intersect B.
The principle takes its name from the action of including the counts of certain sets and then excluding the counts of their intersections to avoid double counting.

The principle can be expanded to any number of sets.
When considering three sets, A, B and C, the principle is: A ∪ B ∪ C = A + B + C - A ∩ B - A ∩ C - B ∩ C + A ∩ B ∩ C .
Here again the sizes of the sets are added, the sizes of all pairs of sets are subtracted, but then the size of the intersection of all three sets is added back.
This pattern continues for larger numbers of sets, alternating between inclusion and exclusion as necessary to count each intersection exactly once.

The principle is frequently used to solve problems in probability and statistics.
It enables the precise calculation of probabilities by considering complementary events.
It is also used in computer science for counting problems and problems involving the manipulation of data sets.

The key to successfully applying The Inclusion-Exclusion Principle is being systematic about the calculation of intersections, ensuring each is counted only once.
Understanding the abstraction of this principle and applying it in different scenarios is essential.