If \(n(U)=100\), \(n(A^c)=40\), \(n(B^c)=55\), and \(n(A^c\cap B^c)=20\), what is \(n(A\cap B)\)?
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Answer and explanation
Correct answer: 25
First apply the inclusion–exclusion formula to the complements: \(n(A^c\cup B^c)=n(A^c)+n(B^c)-n(A^c\cap B^c)=40+55-20=75\). By De Morgan’s law, \(A^c\cup B^c=(A\cap B)^c\). Hence \(n((A\cap B)^c)=75\), and therefore \(n(A\cap B)=n(U)-75=100-75=25\).
Frequently asked questions
What is the correct answer to this question?
25
Why is this the correct answer?
First apply the inclusion–exclusion formula to the complements: \(n(A^c\cup B^c)=n(A^c)+n(B^c)-n(A^c\cap B^c)=40+55-20=75\). By De Morgan’s law, \(A^c\cup B^c=(A\cap B)^c\). Hence \(n((A\cap B)^c)=75\), and therefore \(n(A\cap B)=n(U)-75=100-75=25\).
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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