If |A| = 30, |B| = 27, |C| = 25, |A ∩ B| = 12, |B ∩ C| = 10, |C ∩ A| = 8, and |A ∩ B ∩ C| = 4, what is |A ∪ B ∪ C|?
Answer and explanation
Correct answer: 56
Use the inclusion–exclusion formula for three sets: |A ∪ B ∪ C| = |A| + |B| + |C| − |A ∩ B| − |B ∩ C| − |C ∩ A| + |A ∩ B ∩ C|. Substitution gives 30 + 27 + 25 − 12 − 10 − 8 + 4 = 56. The triple intersection is added back because it was subtracted three times but should be counted once. Hence option A is correct.
Frequently asked questions
What is the correct answer to this question?
56
Why is this the correct answer?
Use the inclusion–exclusion formula for three sets: |A ∪ B ∪ C| = |A| + |B| + |C| − |A ∩ B| − |B ∩ C| − |C ∩ A| + |A ∩ B ∩ C|. Substitution gives 30 + 27 + 25 − 12 − 10 − 8 + 4 = 56. The triple intersection is added back because it was subtracted three times but should be counted once. Hence option A is correct.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Operations on Sets (Union, Intersection, Difference).
