If n(U) = 80, n(A) = 35, n(B) = 42, and n(A ∩ B) = 18, find n((A ∪ B)′).
Answer and explanation
Correct answer: 21
First calculate the number of elements in the union using the inclusion–exclusion formula: n(A ∪ B) = n(A) + n(B) − n(A ∩ B). Substitution gives n(A ∪ B) = 35 + 42 − 18 = 59. The universal set has 80 elements, so the complement of the union contains the elements outside both A and B. Therefore, n((A ∪ B)′) = n(U) − n(A ∪ B) = 80 − 59 = 21. Thus option A is correct.
Frequently asked questions
What is the correct answer to this question?
21
Why is this the correct answer?
First calculate the number of elements in the union using the inclusion–exclusion formula: n(A ∪ B) = n(A) + n(B) − n(A ∩ B). Substitution gives n(A ∪ B) = 35 + 42 − 18 = 59. The universal set has 80 elements, so the complement of the union contains the elements outside both A and B. Therefore, n((A ∪ B)′) = n(U) − n(A ∪ B) = 80 − 59 = 21. Thus option A is correct.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Complement of a Set and Its Properties.
