If A ∩ B = U, what is A′ ∪ B′?
Answer and explanation
Correct answer: ∅
The governing concept is De Morgan’s law: A′ ∪ B′ = (A ∩ B)′. Since A ∩ B = U, taking the complement relative to U gives (A ∩ B)′ = U′ = ∅. Equivalently, if the intersection of A and B is the entire universal set, then both A and B equal U, so each complement is empty and their union is also empty. Therefore option A is correct; U and A ∪ B are not empty in general, while option D is an intersection rather than the required union.
Frequently asked questions
What is the correct answer to this question?
∅
Why is this the correct answer?
The governing concept is De Morgan’s law: A′ ∪ B′ = (A ∩ B)′. Since A ∩ B = U, taking the complement relative to U gives (A ∩ B)′ = U′ = ∅. Equivalently, if the intersection of A and B is the entire universal set, then both A and B equal U, so each complement is empty and their union is also empty. Therefore option A is correct; U and A ∪ B are not empty in general, while option D is an intersection rather than the required union.
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.
