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:

• For 2 finite sets and , which are subsets of Universal set, then and are disjoint sets.

Hence it can be said that,

.

• Similarily for 3 finite sets , and ,

Principle :

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,

Properties :

1. Computes the total number of elements that satisfy at least one of several properties.
2. It prevents the problem of double counting.

Example 1:
As shown in the diagram, 3 finite sets A, B and C with their corresponding values are given. Compute .

Solution :
The values of the corresponding regions, as can be noted from the diagram are –

By applying Inclusion-Exclusion principle,

Applications :

• Derangements
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 .
• Applying the Inclusion-Exclusion principle to i general events and rearranging we get the formula,