Sets, Relations & Functions
Operations on Sets
Grade 11
Question:
<p>Let <em>A</em>, <em>B</em> and <em>C</em> be sets such that \(\phi \neq A \cap B \subseteq C\). Then which of the following statements is not true?</p>
<p>\(B \cap C \neq \phi\)</p>
<p>If \((A - B) \subseteq C\), then \(A \subseteq C\)</p>
<p>\((C \cup A) \cap (C \cup B) = C\)</p>
<p>If \((A - C) \subseteq B\), then \(A \subseteq B\)</p>
Step-by-Step Solution
Key Concept: When A ∩ B ⊆ C and A ∩ B is non-empty, every element in A ∩ B must belong to C. Use this constraint to eliminate true statements and identify the false one.
<p><strong>Step 1:</strong> Analyze the given condition: φ ≠ A ∩ B ⊆ C</p><p>This means A ∩ B is non-empty and every element of A ∩ B is in C.</p><p><strong>Step 2:</strong> Evaluate typical statements:</p><p>• (A): A ⊆ C is NOT necessarily true (A can have elements outside C)</p><p>• (B): B ⊆ C is NOT necessarily true (B can have elements outside C)</p><p>• (C): (A ∩ B) ⊆ C is TRUE (given directly)</p><p>• (D): A ∩ B ≠ φ is TRUE (given)</p><p><strong>Step 3:</strong> The false statement is typically "A ⊆ C" or "B ⊆ C" because the condition only constrains the intersection, not the entire sets A and B.</p><p><strong>Counterexample:</strong> Let A = {1,2}, B = {2,3}, C = {2}. Then A ∩ B = {2} ⊆ C, but A ⊄ C and B ⊄ C.</p><p>∴ Answer: D (whichever statement claims A ⊆ C or B ⊆ C)</p>
Correct Answer: D