If \(|U|=180\), \(|A|=85\), \(|B'|=110\), and \(|A\cap B|=28\), what is the value of \(|(A\cup B)'|\)?
Answer and explanation
Correct answer: 53
Since \(|B'|=110\) in a universal set of 180 elements, \(|B|=|U|-|B'|=180-110=70\). Now apply inclusion-exclusion: \(|A\cup B|=|A|+|B|-|A\cap B|=85+70-28=127\). The complement of the union contains all elements of U that are in neither A nor B, so \(|(A\cup B)'|=|U|-|A\cup B|=180-127=53\). Therefore, option C is correct. The intersection must be subtracted once because its elements were counted in both |A| and |B|.
Frequently asked questions
What is the correct answer to this question?
53
Why is this the correct answer?
Since \(|B'|=110\) in a universal set of 180 elements, \(|B|=|U|-|B'|=180-110=70\). Now apply inclusion-exclusion: \(|A\cup B|=|A|+|B|-|A\cap B|=85+70-28=127\). The complement of the union contains all elements of U that are in neither A nor B, so \(|(A\cup B)'|=|U|-|A\cup B|=180-127=53\). Therefore, option C is correct. The intersection must be subtracted once because its elements were counted in both |A| and |B|.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Complement of a Set and Its Properties.
