Let U = {1, 2, …, 30}, A = {x ∈ U | 2 divides x}, and B = {x ∈ U | 3 divides x}. How many elements are in (A ∪ B)′?
Answer and explanation
Correct answer: 10
Among the integers from 1 to 30, 15 are divisible by 2 and 10 are divisible by 3. The numbers divisible by both 2 and 3 are the multiples of 6, and there are 5 of them: 6, 12, 18, 24, and 30. By inclusion–exclusion, n(A ∪ B) = 15 + 10 − 5 = 20. The complement contains the remaining elements of U, so n((A ∪ B)′) = 30 − 20 = 10. Hence option A is correct.
Frequently asked questions
What is the correct answer to this question?
10
Why is this the correct answer?
Among the integers from 1 to 30, 15 are divisible by 2 and 10 are divisible by 3. The numbers divisible by both 2 and 3 are the multiples of 6, and there are 5 of them: 6, 12, 18, 24, and 30. By inclusion–exclusion, n(A ∪ B) = 15 + 10 − 5 = 20. The complement contains the remaining elements of U, so n((A ∪ B)′) = 30 − 20 = 10. Hence 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.
