<p>(JEE Main / AIEEE 2009) If \(A\), \(B\), and \(C\) are three sets such that \(A \cap B = A \cap C\) and \(A \cup B = A \cup C\), then</p>
Step-by-Step Solution
Key Concept: When both A ∩ B = A ∩ C and A ∪ B = A ∪ C hold simultaneously, every element in B must behave identically to corresponding elements in C with respect to A, forcing B = C through set membership analysis.
<p><strong>Step 1:</strong> Take any element x ∈ B. We need to show x ∈ C.</p><p><strong>Step 2:</strong> Since A ∪ B = A ∪ C, we have x ∈ A ∪ B implies x ∈ A ∪ C.</p><p><strong>Step 3:</strong> <u>Case 1:</u> If x ∈ A, then since A ∩ B = A ∩ C, and x ∈ A ∩ B, we get x ∈ A ∩ C, so x ∈ C.</p><p><strong>Step 4:</strong> <u>Case 2:</u> If x ∉ A, then from x ∈ A ∪ B, we must have x ∈ B (which we assumed). From x ∈ A ∪ C and x ∉ A, we must have x ∈ C.</p><p><strong>Step 5:</strong> Thus every element of B is in C, so B ⊆ C. By symmetric argument (starting with y ∈ C), we get C ⊆ B.</p><p><strong>Step 6:</strong> Therefore B = C.</p><p>∴ Answer: <strong>B = C</strong></p>
Correct Answer: C