If \(A\cup B=A\cup C\) and \(A\cap B=A\cap C\), which conclusion must be true?
Answer and explanation
Correct answer: \(B=C\)
To prove \(B=C\), consider any element \(x\). If \(x\in A\), equality of intersections gives \(x\in B\) exactly when \(x\in C\). If \(x\notin A\), equality of unions gives the same equivalence, because membership in either union must then come from \(B\) or \(C\). Thus every element has identical membership in \(B\) and \(C\), so \(B=C\).
Frequently asked questions
What is the correct answer to this question?
\(B=C\)
Why is this the correct answer?
To prove \(B=C\), consider any element \(x\). If \(x\in A\), equality of intersections gives \(x\in B\) exactly when \(x\in C\). If \(x\notin A\), equality of unions gives the same equivalence, because membership in either union must then come from \(B\) or \(C\). Thus every element has identical membership in \(B\) and \(C\), so \(B=C\).
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Operations on Sets (Union, Intersection, Difference).
