If \(A\), \(B\), and \(C\) are finite sets and \(A\cap B\cap C=\varnothing\), which formula for \(|A\cup B\cup C|\) is correct?
Answer and explanation
Correct answer: \(|A|+|B|+|C|-|A\cap B|-|B\cap C|-|C\cap A|\)
The inclusion–exclusion formula for three finite sets is \(|A\cup B\cup C|=|A|+|B|+|C|-|A\cap B|-|B\cap C|-|C\cap A|+|A\cap B\cap C|\). Since the triple intersection is empty, its cardinality is zero, so the final term contributes nothing. Therefore, the correct formula is option A. Pairwise intersections must be subtracted because their elements were counted twice in the initial sum.
Frequently asked questions
What is the correct answer to this question?
\(|A|+|B|+|C|-|A\cap B|-|B\cap C|-|C\cap A|\)
Why is this the correct answer?
The inclusion–exclusion formula for three finite sets is \(|A\cup B\cup C|=|A|+|B|+|C|-|A\cap B|-|B\cap C|-|C\cap A|+|A\cap B\cap C|\). Since the triple intersection is empty, its cardinality is zero, so the final term contributes nothing. Therefore, the correct formula is option A. Pairwise intersections must be subtracted because their elements were counted twice in the initial sum.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Operations on Sets (Union, Intersection, Difference).
