Discussion
If A, B, C are three sets such that A ∪ B = A ∪ C and A ∩ B = A ∩ C, then:
Answer: B
This is a standard property of sets. Consider an element x ∈ B. Then x ∈ A ∪ B. Since A ∪ B = A ∪ C, we have x ∈ A ∪ C. So, x ∈ A or x ∈ C.
Case 1: x ∈ A. Then x ∈ A and x ∈ B, so x ∈ A ∩ B. Since A ∩ B = A ∩ C, we have x ∈ A ∩ C, which implies x ∈ C.
Case 2: x ∉ A. Since x ∈ A ∪ C, we must have x ∈ C.
In both cases, if x ∈ B, then x ∈ C. So, B ⊂ C. Similarly, we can prove that C ⊂ B. Therefore, B = C.
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