If \(A\setminus B=\varnothing\) and \(B\setminus A=\varnothing\), what is the relation between \(A\) and \(B\)?
Answer and explanation
Correct answer: \(A=B\)
The condition \(A\setminus B=\varnothing\) says that A contains no element outside B, so every element of A belongs to B; hence \(A\subseteq B\). Similarly, \(B\setminus A=\varnothing\) implies \(B\subseteq A\). When each set is a subset of the other, the two sets have exactly the same elements. Therefore \(A=B\), not a proper-subset relation and not necessarily disjoint.
Frequently asked questions
What is the correct answer to this question?
\(A=B\)
Why is this the correct answer?
The condition \(A\setminus B=\varnothing\) says that A contains no element outside B, so every element of A belongs to B; hence \(A\subseteq B\). Similarly, \(B\setminus A=\varnothing\) implies \(B\subseteq A\). When each set is a subset of the other, the two sets have exactly the same elements. Therefore \(A=B\), not a proper-subset relation and not necessarily disjoint.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Operations on Sets (Union, Intersection, Difference).
