If U = {1, 2, ..., 77}, A = {x ∈ U : x is divisible by 7}, and B = {x ∈ U : x is divisible by 11}, what is |A' ∩ B'|?
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Answer and explanation
Correct answer: 60
There are floor(77/7) = 11 multiples of 7 and floor(77/11) = 7 multiples of 11. The common multiples are multiples of lcm(7, 11) = 77, so only 77 is common and |A ∩ B| = 1. Hence |A ∪ B| = 11 + 7 − 1 = 17. By De Morgan’s law, A' ∩ B' = (A ∪ B)', so its size is 77 − 17 = 60. Option A is correct.
Frequently asked questions
What is the correct answer to this question?
60
Why is this the correct answer?
There are floor(77/7) = 11 multiples of 7 and floor(77/11) = 7 multiples of 11. The common multiples are multiples of lcm(7, 11) = 77, so only 77 is common and |A ∩ B| = 1. Hence |A ∪ B| = 11 + 7 − 1 = 17. By De Morgan’s law, A' ∩ B' = (A ∪ B)', so its size is 77 − 17 = 60. 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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