If |A| = 12, |B| = 18, |C| = 20, |A ∩ B| = 5, |B ∩ C| = 7, |C ∩ A| = 4, and |A ∩ B ∩ C| = 2, what is |A ∪ B ∪ C|?
Answer and explanation
Correct answer: 36
Use the inclusion–exclusion formula for three finite sets: |A ∪ B ∪ C| = |A| + |B| + |C| − |A ∩ B| − |B ∩ C| − |C ∩ A| + |A ∩ B ∩ C|. Substitution gives 12 + 18 + 20 − 5 − 7 − 4 + 2 = 36. The pairwise intersections are subtracted because they were counted twice, and the triple intersection is added once because it was then removed too many times.
Frequently asked questions
What is the correct answer to this question?
36
Why is this the correct answer?
Use the inclusion–exclusion formula for three finite sets: |A ∪ B ∪ C| = |A| + |B| + |C| − |A ∩ B| − |B ∩ C| − |C ∩ A| + |A ∩ B ∩ C|. Substitution gives 12 + 18 + 20 − 5 − 7 − 4 + 2 = 36. The pairwise intersections are subtracted because they were counted twice, and the triple intersection is added once because it was then removed too many times.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Operations on Sets (Union, Intersection, Difference).
