If A ∩ B = ∅ and A ∪ B = A ∪ C, which conclusion must be true?
Answer and explanation
Correct answer: B \ C ⊆ A
Take any element x ∈ B \ C. Since x ∈ B, it belongs to A ∪ B. The given equality then places x in A ∪ C. But x ∉ C by the definition of B \ C. Therefore x must belong to A. Since every element of B \ C lies in A, we conclude B \ C ⊆ A. The disjointness condition A ∩ B = ∅ is not needed for this particular conclusion, though it is compatible with the data.
Frequently asked questions
What is the correct answer to this question?
B \ C ⊆ A
Why is this the correct answer?
Take any element x ∈ B \ C. Since x ∈ B, it belongs to A ∪ B. The given equality then places x in A ∪ C. But x ∉ C by the definition of B \ C. Therefore x must belong to A. Since every element of B \ C lies in A, we conclude B \ C ⊆ A. The disjointness condition A ∩ B = ∅ is not needed for this particular conclusion, though it is compatible with the data.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Operations on Sets (Union, Intersection, Difference).
