If n(U) = 90, n(A) = 58, and n(B) = 45, what is the minimum possible value of n(A ∩ B)?
Answer and explanation
Correct answer: 13
The two sets contain 58 + 45 = 103 memberships, but the universal set has only 90 elements. Therefore, at least 103 − 90 = 13 memberships must be repeated in both sets. The general lower bound is n(A ∩ B) ≥ n(A) + n(B) − n(U), so the minimum possible intersection is 58 + 45 − 90 = 13. This value is feasible when the union contains all 90 universal elements.
Frequently asked questions
What is the correct answer to this question?
13
Why is this the correct answer?
The two sets contain 58 + 45 = 103 memberships, but the universal set has only 90 elements. Therefore, at least 103 − 90 = 13 memberships must be repeated in both sets. The general lower bound is n(A ∩ B) ≥ n(A) + n(B) − n(U), so the minimum possible intersection is 58 + 45 − 90 = 13. This value is feasible when the union contains all 90 universal elements.
Which subject and chapter does this question cover?
This is a Class 12 Mathematics question. Chapter: Sets.
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