Let U = {x ∈ ℕ : x ≤ 90}, A = {x ∈ U : 10 divides x}, and B = {x ∈ U : 18 divides x}. What is n(A′ ∩ B′)?
Answer and explanation
Correct answer: 77
By De Morgan’s law, A′ ∩ B′ = (A ∪ B)′. In U, the multiples of 10 are 10, 20, ..., 90, so there are 9; the multiples of 18 are 18, 36, 54, 72, 90, so there are 5. Their only common element is 90 because lcm(10,18) = 90. Thus n(A ∪ B) = 9 + 5 - 1 = 13, and n(A′ ∩ B′) = 90 - 13 = 77. Therefore, option A is correct.
Frequently asked questions
What is the correct answer to this question?
77
Why is this the correct answer?
By De Morgan’s law, A′ ∩ B′ = (A ∪ B)′. In U, the multiples of 10 are 10, 20, ..., 90, so there are 9; the multiples of 18 are 18, 36, 54, 72, 90, so there are 5. Their only common element is 90 because lcm(10,18) = 90. Thus n(A ∪ B) = 9 + 5 - 1 = 13, and n(A′ ∩ B′) = 90 - 13 = 77. Therefore, 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.
