If \(U=\{1,2,\ldots,20\}\), \(A=\{x:x\in U,\ x\text{ is even}\}\), and \(B=\{x:x\in U,\ x\text{ is divisible by }3\}\), then what is \(n((A\cup B)')\)?
Answer and explanation
Correct answer: 7
The even elements of U are \(A=\{2,4,6,8,10,12,14,16,18,20\}\), so \(n(A)=10\). The multiples of 3 are \(B=\{3,6,9,12,15,18\}\), so \(n(B)=6\). Their common elements are \(\{6,12,18\}\), hence \(n(A\cap B)=3\). By inclusion-exclusion, \(n(A\cup B)=10+6-3=13\). Therefore, the complement has \(20-13=7\) elements, so option A is correct.
Frequently asked questions
What is the correct answer to this question?
7
Why is this the correct answer?
The even elements of U are \(A=\{2,4,6,8,10,12,14,16,18,20\}\), so \(n(A)=10\). The multiples of 3 are \(B=\{3,6,9,12,15,18\}\), so \(n(B)=6\). Their common elements are \(\{6,12,18\}\), hence \(n(A\cap B)=3\). By inclusion-exclusion, \(n(A\cup B)=10+6-3=13\). Therefore, the complement has \(20-13=7\) elements, so option A is correct.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Operations on Sets (Union, Intersection, Difference).
