If \(n(A)=19\), \(n(B)=13\), and \(n(A\cup B)=25\), what is \(n(A\cap B)\)?
Answer and explanation
Correct answer: 7
For two finite sets, the inclusion–exclusion formula is \(n(A\cup B)=n(A)+n(B)-n(A\cap B)\). Therefore, \(n(A\cap B)=19+13-25=7\). The intersection is the part counted in both sets, so it must be subtracted once from the sum of the two set sizes. Hence, option B is correct. The values 5 and 12 result from incorrect arithmetic or using the formula in the wrong direction, while 32 is greater than the union and cannot be the intersection.
Frequently asked questions
What is the correct answer to this question?
7
Why is this the correct answer?
For two finite sets, the inclusion–exclusion formula is \(n(A\cup B)=n(A)+n(B)-n(A\cap B)\). Therefore, \(n(A\cap B)=19+13-25=7\). The intersection is the part counted in both sets, so it must be subtracted once from the sum of the two set sizes. Hence, option B is correct. The values 5 and 12 result from incorrect arithmetic or using the formula in the wrong direction, while 32 is greater than the union and cannot be the intersection.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Operations on Sets (Union, Intersection, Difference).
