If \(n(U)=120\), \(n(A)=70\), \(n(B)=65\), and \(n(A\cap B)=40\), what is the value of \(n(A'\cap B')\)?
Answer and explanation
Correct answer: 25
Use inclusion–exclusion to calculate the size of the union: \(n(A\cup B)=n(A)+n(B)-n(A\cap B)=70+65-40=95\). De Morgan’s law gives \(A'\cap B'=(A\cup B)'\), so the required set consists of elements in the universal set that belong to neither A nor B. Its size is therefore \(n(U)-n(A\cup B)=120-95=25\). Thus option A is correct; 95 is the union size, not the required complement size.
Frequently asked questions
What is the correct answer to this question?
25
Why is this the correct answer?
Use inclusion–exclusion to calculate the size of the union: \(n(A\cup B)=n(A)+n(B)-n(A\cap B)=70+65-40=95\). De Morgan’s law gives \(A'\cap B'=(A\cup B)'\), so the required set consists of elements in the universal set that belong to neither A nor B. Its size is therefore \(n(U)-n(A\cup B)=120-95=25\). Thus option A is correct; 95 is the union size, not the required complement size.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Operations on Sets (Union, Intersection, Difference).
