Let \(U=\{1,2,\ldots,54\}\), \(A=\{x:x\text{ is divisible by }6\}\), and \(B=\{x:x\text{ is divisible by }9\}\). What is \(|A'\cup B'|\)?
Advertisement
Answer and explanation
Correct answer: 51
By De Morgan’s law, \(A'\cup B'=(A\cap B)'\). A number belonging to both \(A\) and \(B\) must be divisible by \(\operatorname{lcm}(6,9)=18\). The multiples of 18 from 1 through 54 are 18, 36, and 54, so \(|A\cap B|=3\). Therefore, \(|A'\cup B'|=54-3=51\).
Frequently asked questions
What is the correct answer to this question?
51
Why is this the correct answer?
By De Morgan’s law, \(A'\cup B'=(A\cap B)'\). A number belonging to both \(A\) and \(B\) must be divisible by \(\operatorname{lcm}(6,9)=18\). The multiples of 18 from 1 through 54 are 18, 36, and 54, so \(|A\cap B|=3\). Therefore, \(|A'\cup B'|=54-3=51\).
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.
Advertisement