If \(A\cup B=A\) and \(A\cap B=B\), which of the following statements must be true?
Answer and explanation
Correct answer: \(B\subseteq A\)
The equation \(A\cap B=B\) says that every element of \(B\) is also an element of \(A\), so \(B\subseteq A\). The equation \(A\cup B=A\) gives exactly the same conclusion: adding all elements of \(B\) to \(A\) does not enlarge \(A\). Thus option A is necessary. The reverse inclusion, disjointness, and complement relation do not necessarily follow.
Frequently asked questions
What is the correct answer to this question?
\(B\subseteq A\)
Why is this the correct answer?
The equation \(A\cap B=B\) says that every element of \(B\) is also an element of \(A\), so \(B\subseteq A\). The equation \(A\cup B=A\) gives exactly the same conclusion: adding all elements of \(B\) to \(A\) does not enlarge \(A\). Thus option A is necessary. The reverse inclusion, disjointness, and complement relation do not necessarily follow.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Operations on Sets (Union, Intersection, Difference).
