If A ∩ B ∩ C = ∅, n(A) = 30, n(B) = 34, n(C) = 36, n(A ∩ B) = 8, n(B ∩ C) = 10, and n(C ∩ A) = 6, what is n(A ∪ B ∪ C)?
Advertisement
Answer and explanation
Correct answer: 76
For three sets, inclusion–exclusion gives n(A ∪ B ∪ C) = n(A)+n(B)+n(C)−n(A∩B)−n(B∩C)−n(C∩A)+n(A∩B∩C). The triple intersection is empty, so its cardinality is 0. Substitution gives 30+34+36−8−10−6+0 = 76. Therefore, option A is correct; simply adding all set sizes would double-count pairwise overlaps.
Frequently asked questions
What is the correct answer to this question?
76
Why is this the correct answer?
For three sets, inclusion–exclusion gives n(A ∪ B ∪ C) = n(A)+n(B)+n(C)−n(A∩B)−n(B∩C)−n(C∩A)+n(A∩B∩C). The triple intersection is empty, so its cardinality is 0. Substitution gives 30+34+36−8−10−6+0 = 76. Therefore, option A is correct; simply adding all set sizes would double-count pairwise overlaps.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Operations on Sets (Union, Intersection, Difference).
Advertisement