Sets, Relations & Functions
Set operations
Grade 11
Question:
<p>Let <em>U</em> be the universal set and \(A \cup B \cup C = U\). Then \([(A - B) \cup (B - C) \cup (C - A)]'\) equals</p>
<p>\(A \cup B \cup C\)</p>
<p>\(A \cap B \cap C\)</p>
<p>\(A \cup (B \cap C)\)</p>
<p>\(A \cap (B \cup C)\)</p>
Step-by-Step Solution
Key Concept: The complement of a union equals the intersection of complements (De Morgan's Law). The set (A-B)∪(B-C)∪(C-A) contains elements in exactly one or exactly two of the sets A,B,C. Its complement contains elements in either all three sets or in none of the sets.
<p><strong>Step 1:</strong> Understand what (A-B)∪(B-C)∪(C-A) represents.</p><p>This set contains all elements that belong to exactly one set or exactly two sets among A, B, C (but not all three).</p><p><strong>Step 2:</strong> Find the complement. [(A-B)∪(B-C)∪(C-A)]' contains elements NOT in (A-B)∪(B-C)∪(C-A).</p><p>These are elements that either:</p><p>• Belong to all three sets: A∩B∩C</p><p>• Belong to none of the sets: (A∪B∪C)' = U' = ∅ (since A∪B∪C = U)</p><p><strong>Step 3:</strong> Therefore, [(A-B)∪(B-C)∪(C-A)]' = (A∩B∩C) ∪ ∅ = <strong>A∩B∩C</strong></p><p>∴ Answer: B</p>
Correct Answer: B