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,
.
- Similarly 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 :
- Computes the total number of elements that satisfy at least one of several properties.
- 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 areand 3 1 2.