Let U = {1, 2, ..., 54}, A = {x ∈ U : 9 divides x}, and B = {x ∈ U : 6 divides x}. What is n(A′ ∩ B′)?
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Answer and explanation
Correct answer: 42
By De Morgan’s law, A′ ∩ B′ = (A ∪ B)′. In U = {1, 2, ..., 54}, the number of multiples of 9 is ⌊54/9⌋ = 6, and the number of multiples of 6 is ⌊54/6⌋ = 9. Numbers counted in both groups are multiples of lcm(9, 6) = 18; there are ⌊54/18⌋ = 3. Hence n(A ∪ B) = 6 + 9 − 3 = 12, and n(A′ ∩ B′) = 54 − 12 = 42.
Frequently asked questions
What is the correct answer to this question?
42
Why is this the correct answer?
By De Morgan’s law, A′ ∩ B′ = (A ∪ B)′. In U = {1, 2, ..., 54}, the number of multiples of 9 is ⌊54/9⌋ = 6, and the number of multiples of 6 is ⌊54/6⌋ = 9. Numbers counted in both groups are multiples of lcm(9, 6) = 18; there are ⌊54/18⌋ = 3. Hence n(A ∪ B) = 6 + 9 − 3 = 12, and n(A′ ∩ B′) = 54 − 12 = 42.
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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