If U = {1, 2, …, 120}, and A, B, C are respectively the sets of multiples of 4, 6, and 10, what is |(A ∪ B ∪ C)′|?
Answer and explanation
Correct answer: 76
Use inclusion–exclusion. There are 30 multiples of 4, 20 of 6, and 12 of 10. Pairwise intersections contain 10 multiples of 12, 6 multiples of 20, and 4 multiples of 30; the triple intersection has 2 multiples of 60. Thus |A ∪ B ∪ C| = 30 + 20 + 12 − 10 − 6 − 4 + 2 = 44. Therefore the complement has 120 − 44 = 76 elements, so option C is correct.
Frequently asked questions
What is the correct answer to this question?
76
Why is this the correct answer?
Use inclusion–exclusion. There are 30 multiples of 4, 20 of 6, and 12 of 10. Pairwise intersections contain 10 multiples of 12, 6 multiples of 20, and 4 multiples of 30; the triple intersection has 2 multiples of 60. Thus |A ∪ B ∪ C| = 30 + 20 + 12 − 10 − 6 − 4 + 2 = 44. Therefore the complement has 120 − 44 = 76 elements, 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.
