If n(A ∪ B ∪ C) = 128, 63 elements are in exactly one set, and 11 are in all three sets, how many elements are in exactly two sets?
Answer and explanation
Correct answer: 54
Partition the union into three non-overlapping groups: exactly one set, exactly two sets, and all three sets. Let x be the number in exactly two sets. Then 128 = 63 + x + 11, because each element is counted once in these exclusive categories. Solving gives x = 128 − 63 − 11 = 54. The triple-intersection is subtracted only once; subtracting it twice would incorrectly produce 43.
Frequently asked questions
What is the correct answer to this question?
54
Why is this the correct answer?
Partition the union into three non-overlapping groups: exactly one set, exactly two sets, and all three sets. Let x be the number in exactly two sets. Then 128 = 63 + x + 11, because each element is counted once in these exclusive categories. Solving gives x = 128 − 63 − 11 = 54. The triple-intersection is subtracted only once; subtracting it twice would incorrectly produce 43.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Operations on Sets (Union, Intersection, Difference).
