If \(U=\{1,2,\ldots,40\}\), \(A=\{x:x\in U\text{ and }2\mid x\}\), and \(B=\{x:x\in U\text{ and }5\mid x\}\), what is \(n(A'\cap B')\)?
Answer and explanation
Correct answer: \(16\)
By De Morgan’s law, \(A'\cap B'=(A\cup B)'\). Thus, we count numbers from 1 to 40 that are not divisible by 2 or 5. There are 20 multiples of 2 and 8 multiples of 5, while 4 numbers are multiples of both, namely multiples of 10. Therefore, \(n(A\cup B)=20+8-4=24\), and \(n(A'\cap B')=40-24=16\). Hence, option A is correct.
Frequently asked questions
What is the correct answer to this question?
\(16\)
Why is this the correct answer?
By De Morgan’s law, \(A'\cap B'=(A\cup B)'\). Thus, we count numbers from 1 to 40 that are not divisible by 2 or 5. There are 20 multiples of 2 and 8 multiples of 5, while 4 numbers are multiples of both, namely multiples of 10. Therefore, \(n(A\cup B)=20+8-4=24\), and \(n(A'\cap B')=40-24=16\). Hence, 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.
