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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

  1. Number of families which buy all three newspapers.
  2. Number of families which buy newspaper A only
  3. Number of families which buy newspaper B only
  4. Number of families which buy newspaper C only
  5. Number of families which buy None of A, B, C
  6. Number of families which buy exactly only one newspaper
  7. Number of families which buy newspaper A and B only
  8. Number of families which buy newspaper B and C only
  9. Number of families which buy newspaper C and A only
  10. Number of families which buy at least two newspapers
  11. Number of families which buy at most two newspapers
  12. Number of families which buy exactly two newspapers

Solution:

Inclusion-Exclusion Principle

1. Number of families which buy all three newspapers:

2. Number of families which buy newspaper A only

3. Number of families which buy newspaper B only

4. Number of families which buy newspaper C only

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

7. Number of families which buy newspaper A and B only

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

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

10. Number of families which buy at least two newspapers

11. Number of families which buy at most two newspapers

12. Number of families which buy exactly two newspapers






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