If \(A\cup B=A\cap B\), which of the following conclusions is always true?
Answer and explanation
Correct answer: \(A=B\)
Every element of A belongs to the union \(A\cup B\). Since the union is given to equal \(A\cap B\), every element of A must also belong to B. Thus \(A\subseteq B\). Similarly, every element of B belongs to the union and therefore to A, giving \(B\subseteq A\). Mutual containment proves \(A=B\). The sets need not be empty, so options B and C are not always true, and D is also not necessary.
Frequently asked questions
What is the correct answer to this question?
\(A=B\)
Why is this the correct answer?
Every element of A belongs to the union \(A\cup B\). Since the union is given to equal \(A\cap B\), every element of A must also belong to B. Thus \(A\subseteq B\). Similarly, every element of B belongs to the union and therefore to A, giving \(B\subseteq A\). Mutual containment proves \(A=B\). The sets need not be empty, so options B and C are not always true, and D is also not necessary.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Operations on Sets (Union, Intersection, Difference).
