If A ∩ B = A ∩ C and A ∪ B = A ∪ C, which conclusion is correct?
Answer and explanation
Correct answer: B = C
Consider any element x. If x belongs to A, the equality A ∩ B = A ∩ C forces x to belong to B exactly when it belongs to C. If x does not belong to A, the equality A ∪ B = A ∪ C forces the same conclusion. Thus every element belongs to B and C together or to neither, so B = C. The stronger claim A = B = C does not necessarily follow.
Frequently asked questions
What is the correct answer to this question?
B = C
Why is this the correct answer?
Consider any element x. If x belongs to A, the equality A ∩ B = A ∩ C forces x to belong to B exactly when it belongs to C. If x does not belong to A, the equality A ∪ B = A ∪ C forces the same conclusion. Thus every element belongs to B and C together or to neither, so B = C. The stronger claim A = B = C does 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).
