If the universal set is U = {1, 2, ..., 42}, A = {x ∈ U : x is divisible by 3}, and B = {x ∈ U : x is divisible by 7}, what is |A' ∪ B'|?
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Answer and explanation
Correct answer: 40
By De Morgan’s law, A' ∪ B' = (A ∩ B)'. An element belongs to A ∩ B when it is divisible by both 3 and 7, so it must be divisible by lcm(3, 7) = 21. Between 1 and 42, the multiples of 21 are 21 and 42; hence |A ∩ B| = 2. Therefore, |A' ∪ B'| = |U| − |A ∩ B| = 42 − 2 = 40. Thus option B is correct.
Frequently asked questions
What is the correct answer to this question?
40
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 7, so it must be divisible by lcm(3, 7) = 21. Between 1 and 42, the multiples of 21 are 21 and 42; hence |A ∩ B| = 2. Therefore, |A' ∪ B'| = |U| − |A ∩ B| = 42 − 2 = 40. Thus option B 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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