Inclusion Exclusion Principle

Let A, B be any two finite sets. Then n (A ∪ B) = n (A) + n (B) - n (A ∩ B)

Here "include" n (A) and n (B) and we "exclude" n (A ∩ B)

Example 1: Suppose A, B, C are finite sets. Then A ∪ B ∪ C is finite and n (A ∪ B ∪ C) = n(A) + n(B) + n(C) - n(A ∩ B) - n(A ∩ C) - n(B ∩ C) + n(A ∩ B ∩ C)

Example 2: In a town of 10000 families it was found that 40% of families buy newspaper A, 20% family buy newspaper B, 10% family buy newspaper C, 5% family buy newspaper A and B, 3% family buy newspaper B and C and 4% family buy newspaper A and C. If 2% family buy all the newspaper. Find the number of families which buy

Number of families which buy all three newspapers. Number of families which buy newspaper A only Number of families which buy newspaper B only Number of families which buy newspaper C only Number of families which buy None of A, B, C Number of families which buy exactly only one newspaper Number of families which buy newspaper A and B only Number of families which buy newspaper B and C only Number of families which buy newspaper C and A only Number of families which buy at least two newspapers Number of families which buy at most two newspapers Number of families which buy exactly two newspapers Solution:

Inclusion-Exclusion Principle 1. Number of families which buy all three newspapers:

n (A ∪ B ∪ C) = n(A) + n(B) + n(C) - n(A ∩ B) - n(A ∩ C) - n(B ∩ C) + n(A ∩ B ∩ C)
n (A ∪ B ∪ C) = 40 + 20 + 10 - 5 - 3 - 4 + 2 = 60%
2. Number of families which buy newspaper A only

= 40 - 7 = 33%
3. Number of families which buy newspaper B only

= 20 - 6 = 14%
4. Number of families which buy newspaper C only

= 10 - 5 = 5%
5. Number of families which buy None of A, B, and C

n (A ∪B ∪C)c = 100 - n (A ∪ B ∪ C) n (A ∪B ∪C)c = 100 - [40 + 20 + 10 - 5- 3- 4 + 2] n (A ∪B ∪C)c = 100 - 60 = 40 % 6. Number of families which buy exactly only one newspaper

= 33 + 14 + 5 = 52%
7. Number of families which buy newspaper A and B only

= 3%
8. Number of families which buy newspaper B and C only

= 1%
9. Number of families which buy newspaper C and A only

= 2%
10. Number of families which buy at least two newspapers

= 8%
11. Number of families which buy at most two newspapers

= 98%
12. Number of families which buy exactly two newspapers

= 6%