In a survey, n(U) = 95, n(A) = 48, n(B) = 39, and n((A ∪ B)ᶜ) = 20. According to the Venn diagram, what is n(A ∩ B)?
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Answer and explanation
Correct answer: 12
First find the union from its complement: n(A ∪ B) = n(U) − n((A ∪ B)ᶜ) = 95 − 20 = 75. Then apply inclusion–exclusion, n(A ∪ B) = n(A) + n(B) − n(A ∩ B). Thus 75 = 48 + 39 − n(A ∩ B), so n(A ∩ B) = 87 − 75 = 12. Option A is correct; the complement value 20 is not the intersection.
Frequently asked questions
What is the correct answer to this question?
12
Why is this the correct answer?
First find the union from its complement: n(A ∪ B) = n(U) − n((A ∪ B)ᶜ) = 95 − 20 = 75. Then apply inclusion–exclusion, n(A ∪ B) = n(A) + n(B) − n(A ∩ B). Thus 75 = 48 + 39 − n(A ∩ B), so n(A ∩ B) = 87 − 75 = 12. Option A is correct; the complement value 20 is not the intersection.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Venn Diagrams.
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