Let U = {1, 2, 3, …, 60}. If A = {x ∈ U : 4 divides x} and B = {x ∈ U : 9 divides x}, how many elements are in the complement of A ∪ B with respect to U?
Advertisement
Answer and explanation
Correct answer: 40
There are floor(60/4) = 15 multiples of 4 and floor(60/9) = 6 multiples of 9. Numbers in both sets must be multiples of lcm(4, 9) = 36; only 36 occurs up to 60. Thus n(A ∪ B) = 15 + 6 − 1 = 20 by inclusion–exclusion. The complement contains the remaining 60 − 20 = 40 elements.
Frequently asked questions
What is the correct answer to this question?
40
Why is this the correct answer?
There are floor(60/4) = 15 multiples of 4 and floor(60/9) = 6 multiples of 9. Numbers in both sets must be multiples of lcm(4, 9) = 36; only 36 occurs up to 60. Thus n(A ∪ B) = 15 + 6 − 1 = 20 by inclusion–exclusion. The complement contains the remaining 60 − 20 = 40 elements.
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.
Advertisement