Let U = {1,2,...,55}, A = {x ∈ U : x is divisible by 5}, and B = {x ∈ U : x is divisible by 11}. What is |A' ∩ B'|?
Answer and explanation
Correct answer: 40
By De Morgan's law, A' ∩ B' = (A ∪ B)'. In U, the multiples of 5 are counted by floor(55/5) = 11, and the multiples of 11 by floor(55/11) = 5. The common multiples are multiples of 55; only 55 lies in U, so inclusion–exclusion gives |A ∪ B| = 11 + 5 − 1 = 15. Therefore |A' ∩ B'| = 55 − 15 = 40, so option C 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)'. In U, the multiples of 5 are counted by floor(55/5) = 11, and the multiples of 11 by floor(55/11) = 5. The common multiples are multiples of 55; only 55 lies in U, so inclusion–exclusion gives |A ∪ B| = 11 + 5 − 1 = 15. Therefore |A' ∩ B'| = 55 − 15 = 40, so option C 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.
