If U = {1, 2, ..., 30}, A is the set of prime numbers and B is the set of multiples of 3, then how many elements are in (A ∪ B)'?
Answer and explanation
Correct answer: 11
The prime numbers in U are 2, 3, 5, 7, 11, 13, 17, 19, 23 and 29, so |A| = 10. The multiples of 3 from 1 to 30 are 3, 6, 9, 12, 15, 18, 21, 24, 27 and 30, so |B| = 10. Their only common element is 3, hence |A ∩ B| = 1. By inclusion-exclusion, |A ∪ B| = 10 + 10 − 1 = 19. Therefore, |(A ∪ B)'| = |U| − |A ∪ B| = 30 − 19 = 11. Thus, option A is correct.
Frequently asked questions
What is the correct answer to this question?
11
Why is this the correct answer?
The prime numbers in U are 2, 3, 5, 7, 11, 13, 17, 19, 23 and 29, so |A| = 10. The multiples of 3 from 1 to 30 are 3, 6, 9, 12, 15, 18, 21, 24, 27 and 30, so |B| = 10. Their only common element is 3, hence |A ∩ B| = 1. By inclusion-exclusion, |A ∪ B| = 10 + 10 − 1 = 19. Therefore, |(A ∪ B)'| = |U| − |A ∪ B| = 30 − 19 = 11. Thus, 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.
