If \(U=\{1,2,\ldots,60\}\), \(A=\{x:x\in U,2\mid x\}\), and \(B=\{x:x\in U,3\mid x\}\), what is \(n(A'\cap B')\)?
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Answer and explanation
Correct answer: 20
By De Morgan's law, \(A'\cap B'=(A\cup B)'\), so we count numbers from 1 to 60 divisible by neither 2 nor 3. There are 30 multiples of 2, 20 multiples of 3, and 10 multiples of both 2 and 3. Thus \(n(A\cup B)=30+20-10=40\). Therefore, \(n(A'\cap B')=60-40=20\), so option A is correct.
Frequently asked questions
What is the correct answer to this question?
20
Why is this the correct answer?
By De Morgan's law, \(A'\cap B'=(A\cup B)'\), so we count numbers from 1 to 60 divisible by neither 2 nor 3. There are 30 multiples of 2, 20 multiples of 3, and 10 multiples of both 2 and 3. Thus \(n(A\cup B)=30+20-10=40\). Therefore, \(n(A'\cap B')=60-40=20\), so option A 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.
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