If n(U) = 90, n(A) = 58, and n(B) = 49, what is the minimum possible value of n(A ∩ B)?
Answer and explanation
Correct answer: 17
For two subsets of a universal set, n(A ∪ B) = n(A) + n(B) − n(A ∩ B), and n(A ∪ B) cannot exceed n(U). To make the intersection as small as possible, the union must be as large as possible, namely 90. Hence 90 ≥ 58 + 49 − n(A ∩ B), which gives n(A ∩ B) ≥ 17. This bound is attainable, so the minimum is 17.
Frequently asked questions
What is the correct answer to this question?
17
Why is this the correct answer?
For two subsets of a universal set, n(A ∪ B) = n(A) + n(B) − n(A ∩ B), and n(A ∪ B) cannot exceed n(U). To make the intersection as small as possible, the union must be as large as possible, namely 90. Hence 90 ≥ 58 + 49 − n(A ∩ B), which gives n(A ∩ B) ≥ 17. This bound is attainable, so the minimum is 17.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Operations on Sets (Union, Intersection, Difference).
