If U = {1, 2, ..., 90}, and A, B, C are respectively the sets of multiples of 2, 3, and 5, what is |(A ∪ B ∪ C)'|?
Answer and explanation
Correct answer: 24
Use inclusion–exclusion. There are 45 multiples of 2, 30 of 3, and 18 of 5. Pairwise overlaps contain 15 multiples of 6, 9 of 10, and 6 of 15; the triple overlap contains 3 multiples of 30. Thus |A ∪ B ∪ C| = 45 + 30 + 18 − 15 − 9 − 6 + 3 = 66. Therefore, the complement has 90 − 66 = 24 elements, so option B is correct.
Frequently asked questions
What is the correct answer to this question?
24
Why is this the correct answer?
Use inclusion–exclusion. There are 45 multiples of 2, 30 of 3, and 18 of 5. Pairwise overlaps contain 15 multiples of 6, 9 of 10, and 6 of 15; the triple overlap contains 3 multiples of 30. Thus |A ∪ B ∪ C| = 45 + 30 + 18 − 15 − 9 − 6 + 3 = 66. Therefore, the complement has 90 − 66 = 24 elements, so 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.
