If \(U=\{1,2,\ldots,40\}\), \(A=\{x:x\in U,\ 2\mid x\}\), \(B=\{x:x\in U,\ 3\mid x\}\), and \(C=\{x:x\in U,\ 5\mid x\}\), then what is \(n(A\cup B\cup C)\)?
Answer and explanation
Correct answer: 30
There are \(20\) multiples of 2, \(13\) multiples of 3, and \(8\) multiples of 5 in \(\{1,\ldots,40\}\). The pairwise overlaps contain multiples of 6, 10, and 15, giving \(6,4,2\) elements respectively. The common overlap consists of multiples of 30, so it has 1 element. Therefore, by inclusion–exclusion, \(n(A\cup B\cup C)=20+13+8-6-4-2+1=30\).
Frequently asked questions
What is the correct answer to this question?
30
Why is this the correct answer?
There are \(20\) multiples of 2, \(13\) multiples of 3, and \(8\) multiples of 5 in \(\{1,\ldots,40\}\). The pairwise overlaps contain multiples of 6, 10, and 15, giving \(6,4,2\) elements respectively. The common overlap consists of multiples of 30, so it has 1 element. Therefore, by inclusion–exclusion, \(n(A\cup B\cup C)=20+13+8-6-4-2+1=30\).
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Operations on Sets (Union, Intersection, Difference).
