If U = {1, 2, ..., 45}, A = {x ∈ U : 3 divides x}, and B = {x ∈ U : 5 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)′. An element belongs to A ∩ B when it is divisible by both 3 and 5, so it must be divisible by 15. The multiples of 15 in U are 15, 30, and 45, giving n(A ∩ B) = 3. Since U has 45 elements, n((A ∩ B)′) = 45 − 3 = 42. Therefore, option A is correct.
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)′. An element belongs to A ∩ B when it is divisible by both 3 and 5, so it must be divisible by 15. The multiples of 15 in U are 15, 30, and 45, giving n(A ∩ B) = 3. Since U has 45 elements, n((A ∩ B)′) = 45 − 3 = 42. 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.
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