If n(U) = 180, n(A) = 105, and n(B) = 96, what is the minimum possible value of n(A ∩ B)?
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Answer and explanation
Correct answer: 21
For two subsets of a universal set, n(A ∪ B) cannot exceed n(U). Since n(A ∪ B) = n(A) + n(B) − n(A ∩ B), the minimum intersection occurs when the union is as large as possible, namely 180. Therefore, n(A ∩ B) = 105 + 96 − 180 = 21. Thus, at least 21 elements must be common to A and B.
Frequently asked questions
What is the correct answer to this question?
21
Why is this the correct answer?
For two subsets of a universal set, n(A ∪ B) cannot exceed n(U). Since n(A ∪ B) = n(A) + n(B) − n(A ∩ B), the minimum intersection occurs when the union is as large as possible, namely 180. Therefore, n(A ∩ B) = 105 + 96 − 180 = 21. Thus, at least 21 elements must be common to A and B.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Venn Diagrams.
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