If U = {x : x ∈ ℕ, x ≤ 96}, A = {x : x ∈ U, 8 divides x}, and B = {x : x ∈ U, 12 divides x}, what is n(A′ ∩ B′)?
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Answer and explanation
Correct answer: 80
Using De Morgan’s law, A′ ∩ B′ = (A ∪ B)′. There are 96/8 = 12 multiples of 8 and 96/12 = 8 multiples of 12 in U. Numbers counted in both sets are multiples of lcm(8,12) = 24; there are 96/24 = 4 of them. Thus n(A ∪ B) = 12 + 8 − 4 = 16. Since U has 96 elements, n(A′ ∩ B′) = 96 − 16 = 80.
Frequently asked questions
What is the correct answer to this question?
80
Why is this the correct answer?
Using De Morgan’s law, A′ ∩ B′ = (A ∪ B)′. There are 96/8 = 12 multiples of 8 and 96/12 = 8 multiples of 12 in U. Numbers counted in both sets are multiples of lcm(8,12) = 24; there are 96/24 = 4 of them. Thus n(A ∪ B) = 12 + 8 − 4 = 16. Since U has 96 elements, n(A′ ∩ B′) = 96 − 16 = 80.
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.
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