If U = {x : x ∈ ℕ, x ≤ 84}, A = {x : x ∈ U, 6 divides x}, and B = {x : x ∈ U, 14 divides x}, what is n(A′ ∩ B′)?
Answer and explanation
Correct answer: 66
By De Morgan’s law, A′ ∩ B′ = (A ∪ B)′. The set A contains the multiples of 6 up to 84, so n(A) = 84/6 = 14. The set B contains the multiples of 14, so n(B) = 84/14 = 6. Their common elements are multiples of lcm(6,14) = 42, giving n(A ∩ B) = 84/42 = 2. Therefore n(A ∪ B) = 14 + 6 − 2 = 18, and n(A′ ∩ B′) = 84 − 18 = 66.
Frequently asked questions
What is the correct answer to this question?
66
Why is this the correct answer?
By De Morgan’s law, A′ ∩ B′ = (A ∪ B)′. The set A contains the multiples of 6 up to 84, so n(A) = 84/6 = 14. The set B contains the multiples of 14, so n(B) = 84/14 = 6. Their common elements are multiples of lcm(6,14) = 42, giving n(A ∩ B) = 84/42 = 2. Therefore n(A ∪ B) = 14 + 6 − 2 = 18, and n(A′ ∩ B′) = 84 − 18 = 66.
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.
