If \(U=\{1,2,\ldots,60\}\), \(A\) is the set of multiples of 3, \(B\) the set of multiples of 4, and \(C\) the set of multiples of 5, what is \(|(A\cup B\cup C)'|\)?
Answer and explanation
Correct answer: 24
There are 20 multiples of 3, 15 of 4, and 12 of 5 in the universal set. Their pairwise overlaps contain 5 multiples of 12, 4 of 15, and 3 of 20; the triple overlap contains 1 multiple of 60. Inclusion-exclusion gives \(|A\cup B\cup C|=20+15+12-5-4-3+1=36\). Therefore, the complement has \(60-36=24\) elements, so C is correct.
Frequently asked questions
What is the correct answer to this question?
24
Why is this the correct answer?
There are 20 multiples of 3, 15 of 4, and 12 of 5 in the universal set. Their pairwise overlaps contain 5 multiples of 12, 4 of 15, and 3 of 20; the triple overlap contains 1 multiple of 60. Inclusion-exclusion gives \(|A\cup B\cup C|=20+15+12-5-4-3+1=36\). Therefore, the complement has \(60-36=24\) elements, so C 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.
