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CMA Foundation · Fundamentals of Business Mathematics and Statistics · Set Theory, including Venn Diagram

For sets A and B in a universal set U with n(U) = 120, it is given that n(A′ ∪ B′) = 90 and n(A) = 50. Using De Morgan's law, what is n(A ∩ B) and hence n(A ∩ B′)?

n(A ∩ B) is 30 and n(A ∩ B′) is 20. De Morgan's law gives A′ ∪ B′ as the complement of A ∩ B, so the intersection is 120 − 90 = 30. Elements of A outside B are then 50 − 30 = 20.

  1. An(A ∩ B) = 30 and n(A ∩ B′) = 20Correct
  2. Bn(A ∩ B) = 90 and n(A ∩ B′) = 0
  3. Cn(A ∩ B) = 30 and n(A ∩ B′) = 80
  4. Dn(A ∩ B) = 20 and n(A ∩ B′) = 30

Explanation

By De Morgan, A′ ∪ B′ = (A ∩ B)′, so n(A ∩ B) = 120 − 90 = 30. Then n(A ∩ B′) = n(A) − n(A ∩ B) = 50 − 30 = 20. The option with 80 wrongly uses 120 − 30 − ... instead of subtracting from n(A).

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