If n(U) = 125, n(A) = 62, n(B) = 55, and n((A ∪ B)ᶜ) = 24, what is n(A ∩ B)?
Answer and explanation
Correct answer: 16
The complement of A ∪ B contains the elements of the universal set that are outside both A and B. Thus n(A ∪ B) = n(U) − n((A ∪ B)ᶜ) = 125 − 24 = 101. Using the inclusion–exclusion formula, n(A ∪ B) = n(A) + n(B) − n(A ∩ B). Therefore, 101 = 62 + 55 − n(A ∩ B), giving n(A ∩ B) = 117 − 101 = 16. Hence option A is correct.
Frequently asked questions
What is the correct answer to this question?
16
Why is this the correct answer?
The complement of A ∪ B contains the elements of the universal set that are outside both A and B. Thus n(A ∪ B) = n(U) − n((A ∪ B)ᶜ) = 125 − 24 = 101. Using the inclusion–exclusion formula, n(A ∪ B) = n(A) + n(B) − n(A ∩ B). Therefore, 101 = 62 + 55 − n(A ∩ B), giving n(A ∩ B) = 117 − 101 = 16. Hence option A is correct.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Operations on Sets (Union, Intersection, Difference).
