Inclusion-Exclusion and its various Applications
In the field of Combinatorics, it is a counting method used to compute the cardinality of the union set. According to basic Inclusion-Exclusion principle:
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- For 2 finite sets and , which are subsets of Universal set, then and are disjoint sets.
Hence it can be said that,
- Similarly for 3 finite sets , and ,
Inclusion-Exclusion principle says that for any number of finite sets , Union of the sets is given by = Sum of sizes of all single sets – Sum of all 2-set intersections + Sum of all the 3-set intersections – Sum of all 4-set intersections .. + Sum of all the i-set intersections.
In general it can be said that,
- Computes the total number of elements that satisfy at least one of several properties.
- It prevents the problem of double counting.
As shown in the diagram, 3 finite sets A, B and C with their corresponding values are given. Compute .
The values of the corresponding regions, as can be noted from the diagram are –
By applying Inclusion-Exclusion principle,
To determine the number of derangements( or permutations) of n objects such that no object is in its original position (like Hat-check problem).
As an example we can consider the derangements of the number in the following cases:
For i = 1, the total number of derangements is 0.
For i = 2, the total number of derangements is 1. This is .
For i = 3, the total number of derangements is 2. These are and 3 1 2.