If n(U) = 75, n(A) = 46, and n(B) = 41, what is the minimum possible value of n(A ∩ B)?
Answer and explanation
Correct answer: 12
The governing principle is the inclusion–exclusion formula: n(A ∪ B) = n(A) + n(B) − n(A ∩ B). Because A ∪ B is contained in U, its size cannot exceed 75. Thus 46 + 41 − n(A ∩ B) ≤ 75, so n(A ∩ B) ≥ 12. This bound is attainable when the union has 75 elements, making 12 the minimum. Zero ignores the limited size of U.
Frequently asked questions
What is the correct answer to this question?
12
Why is this the correct answer?
The governing principle is the inclusion–exclusion formula: n(A ∪ B) = n(A) + n(B) − n(A ∩ B). Because A ∪ B is contained in U, its size cannot exceed 75. Thus 46 + 41 − n(A ∩ B) ≤ 75, so n(A ∩ B) ≥ 12. This bound is attainable when the union has 75 elements, making 12 the minimum. Zero ignores the limited size of U.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Operations on Sets (Union, Intersection, Difference).
