<p>If <em>A</em>, <em>B</em> and <em>C</em> 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 A∩B = A∩C and A∪B = A∪C hold simultaneously, every element in B must equal every corresponding element in C. This is because the intersection condition constrains common elements with A identically, and the union condition ensures no extra elements exist in either B or C outside A.
<p><strong>Step 1:</strong> Assume x ∈ B. We need to show x ∈ C.</p><p><strong>Step 2:</strong> Case 1: If x ∈ A, then x ∈ A∩B. Since A∩B = A∩C, we have x ∈ A∩C, so x ∈ C.</p><p><strong>Step 3:</strong> Case 2: If x ∉ A, then x ∈ B but x ∉ A. This means x ∈ A∪B. Since A∪B = A∪C, we have x ∈ A∪C. Since x ∉ A, we must have x ∈ C.</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> Hence B = C.</p><p>∴ Answer: C (which states B = C)</p>
Correct Answer: C