If \(A\cup B=A\cup C\) and \(A\cap B=A\cap C\), which of the following is correct?
Answer and explanation
Correct answer: \(B=C\)
To prove the result, consider any element x. If x belongs to A, the equality of intersections tells us that x belongs to B exactly when it belongs to C. If x does not belong to A, the equality of unions tells us that x belongs to B exactly when it belongs to C. Thus every element has the same membership status in B and C, so B and C contain precisely the same elements. Therefore \(B=C\). The other options are not forced by the two given equalities.
Frequently asked questions
What is the correct answer to this question?
\(B=C\)
Why is this the correct answer?
To prove the result, consider any element x. If x belongs to A, the equality of intersections tells us that x belongs to B exactly when it belongs to C. If x does not belong to A, the equality of unions tells us that x belongs to B exactly when it belongs to C. Thus every element has the same membership status in B and C, so B and C contain precisely the same elements. Therefore \(B=C\). The other options are not forced by the two given equalities.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Operations on Sets (Union, Intersection, Difference).
