If \(|U|=150\), \(|A'|=92\), \(|B'|=76\), and \(|A\cap B|=31\), what is \(|(A\cup B)'|\)?
Answer and explanation
Correct answer: 49
First convert the complement cardinalities: \(|A|=150-92=58\) and \(|B|=150-76=74\). By inclusion-exclusion, \(|A\cup B|=|A|+|B|-|A\cap B|=58+74-31=101\). Therefore, the complement of the union contains \(|U|-|A\cup B|=150-101=49\) elements. Subtracting the intersection avoids counting common elements twice.
Frequently asked questions
What is the correct answer to this question?
49
Why is this the correct answer?
First convert the complement cardinalities: \(|A|=150-92=58\) and \(|B|=150-76=74\). By inclusion-exclusion, \(|A\cup B|=|A|+|B|-|A\cap B|=58+74-31=101\). Therefore, the complement of the union contains \(|U|-|A\cup B|=150-101=49\) elements. Subtracting the intersection avoids counting common elements twice.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Operations on Sets (Union, Intersection, Difference).
