If U = {1, 2, ..., 90}, A = {x ∈ U : 6 divides x}, and B = {x ∈ U : 15 divides x}, what is |A ∪ B|?
Answer and explanation
Correct answer: 18
The elements of A are the multiples of 6 from 1 to 90, so |A| = floor(90/6) = 15. The elements of B are the multiples of 15, so |B| = floor(90/15) = 6. Elements in both sets are multiples of lcm(6, 15) = 30; there are floor(90/30) = 3 such elements. By inclusion-exclusion, |A ∪ B| = |A| + |B| − |A ∩ B| = 15 + 6 − 3 = 18.
Frequently asked questions
What is the correct answer to this question?
18
Why is this the correct answer?
The elements of A are the multiples of 6 from 1 to 90, so |A| = floor(90/6) = 15. The elements of B are the multiples of 15, so |B| = floor(90/15) = 6. Elements in both sets are multiples of lcm(6, 15) = 30; there are floor(90/30) = 3 such elements. By inclusion-exclusion, |A ∪ B| = |A| + |B| − |A ∩ B| = 15 + 6 − 3 = 18.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Operations on Sets (Union, Intersection, Difference).
