If \(n(U)=120\), \(n(A)=65\), \(n(B)=58\), and \(n(A\cap B)=29\), what is \(n(A^c\cap B^c)\)?
Answer and explanation
Correct answer: 26
By De Morgan’s law, \(A^c\cap B^c=(A\cup B)^c\). First calculate the number of elements in the union: \(n(A\cup B)=n(A)+n(B)-n(A\cap B)=65+58-29=94\). The required region is outside both sets, so subtract the union from the universal set: \(n(A^c\cap B^c)=n(U)-n(A\cup B)=120-94=26\). Therefore, option A is correct. The value 94 represents the union, not its complement, while 29 represents only the intersection.
Frequently asked questions
What is the correct answer to this question?
26
Why is this the correct answer?
By De Morgan’s law, \(A^c\cap B^c=(A\cup B)^c\). First calculate the number of elements in the union: \(n(A\cup B)=n(A)+n(B)-n(A\cap B)=65+58-29=94\). The required region is outside both sets, so subtract the union from the universal set: \(n(A^c\cap B^c)=n(U)-n(A\cup B)=120-94=26\). Therefore, option A is correct. The value 94 represents the union, not its complement, while 29 represents only the intersection.
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.
