If U = {1, 2, ..., 84}, A is the set of multiples of 7 and B is the set of multiples of 4, what is n((A ∪ B)')?
Answer and explanation
Correct answer: 54
The complement of A ∪ B contains elements of U that are neither multiples of 7 nor multiples of 4. There are 84/7 = 12 multiples of 7 and 84/4 = 21 multiples of 4. Their overlap consists of multiples of lcm(7,4) = 28, giving 84/28 = 3 elements. Thus n(A ∪ B) = 12 + 21 − 3 = 30, and n((A ∪ B)') = 84 − 30 = 54. Therefore option A is correct.
Frequently asked questions
What is the correct answer to this question?
54
Why is this the correct answer?
The complement of A ∪ B contains elements of U that are neither multiples of 7 nor multiples of 4. There are 84/7 = 12 multiples of 7 and 84/4 = 21 multiples of 4. Their overlap consists of multiples of lcm(7,4) = 28, giving 84/28 = 3 elements. Thus n(A ∪ B) = 12 + 21 − 3 = 30, and n((A ∪ B)') = 84 − 30 = 54. Therefore option A is correct.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Venn Diagrams.
