If U = {1, 2, ..., 72}, A = {x ∈ U : 4 divides x}, B = {x ∈ U : 6 divides x}, and C = {x ∈ U : 9 divides x}, what is n((A ∪ B ∪ C)')?
Answer and explanation
Correct answer: 44
Use inclusion–exclusion. There are 18 multiples of 4, 12 of 6, and 8 of 9. Pairwise intersections have sizes 6, 2, and 4 because the relevant LCMs are 12, 36, and 18. The triple intersection has size 2, from multiples of 36. Thus n(A ∪ B ∪ C) = 18 + 12 + 8 - 6 - 2 - 4 + 2 = 28. Therefore the complement has 72 - 28 = 44 elements.
Frequently asked questions
What is the correct answer to this question?
44
Why is this the correct answer?
Use inclusion–exclusion. There are 18 multiples of 4, 12 of 6, and 8 of 9. Pairwise intersections have sizes 6, 2, and 4 because the relevant LCMs are 12, 36, and 18. The triple intersection has size 2, from multiples of 36. Thus n(A ∪ B ∪ C) = 18 + 12 + 8 - 6 - 2 - 4 + 2 = 28. Therefore the complement has 72 - 28 = 44 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.
