If \(A\cap B=B\cap C=C\cap A=\varnothing\), \(n(A)=12\), \(n(B)=15\), and \(n(C)=18\), what is the value of \(n(A\cup B\cup C)\)?
Answer and explanation
Correct answer: 45
The conditions \(A\cap B=\varnothing\), \(B\cap C=\varnothing\), and \(C\cap A=\varnothing\) show that the three sets are pairwise disjoint. Therefore, no element is repeated in their union. We can add their cardinalities directly: \(n(A\cup B\cup C)=n(A)+n(B)+n(C)=12+15+18=45\). Hence, option A is correct. If the sets were not disjoint, common elements would have to be subtracted using the inclusion–exclusion principle.
Frequently asked questions
What is the correct answer to this question?
45
Why is this the correct answer?
The conditions \(A\cap B=\varnothing\), \(B\cap C=\varnothing\), and \(C\cap A=\varnothing\) show that the three sets are pairwise disjoint. Therefore, no element is repeated in their union. We can add their cardinalities directly: \(n(A\cup B\cup C)=n(A)+n(B)+n(C)=12+15+18=45\). Hence, option A is correct. If the sets were not disjoint, common elements would have to be subtracted using the inclusion–exclusion principle.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Operations on Sets (Union, Intersection, Difference).
