<p>If \(A \cup B = A \cup C\) and \(A \cap B = A \cap C\), then which of the following is true?</p>
Step-by-Step Solution
Key Concept: Use the absorption property: if two unions are equal and two intersections are equal, then the symmetric difference of B and C with respect to A must be empty, forcing B = C.
<p><strong>Step 1:</strong> Assume any element x ∈ B. We need to show x ∈ C.</p><p><strong>Step 2:</strong> Since x ∈ B, we have x ∈ A ∪ B. By given condition, A ∪ B = A ∪ C, so x ∈ A ∪ C.</p><p><strong>Step 3:</strong> This means x ∈ A or x ∈ C. If x ∈ A, then x ∈ A ∩ B (since x ∈ B). By given condition, A ∩ B = A ∩ C, so x ∈ A ∩ C, which means x ∈ C. If x ∈ C directly, we're done.</p><p><strong>Step 4:</strong> Therefore B ⊆ C. By symmetric argument (starting with x ∈ C), we get C ⊆ B.</p><p><strong>Step 5:</strong> Thus B = C.</p><p>∴ Answer: B = C</p>
Correct Answer: C