Sets, Relations & Functions
Set operations
Grade 11
Question:
<p>The set \((A \cap B')' \cup (B \cap C)\) is equal to</p>
<p>\(A' \cup B \cup C\)</p>
<p>\(A' \cup B\)</p>
<p>\(A' \cup C'\)</p>
<p>\(A' \cap B\)</p>
Step-by-Step Solution
Key Concept: Use De Morgan's Law: (A ∩ B')' = A' ∪ B, then simplify the union with (B ∩ C) by identifying overlapping regions and applying absorption properties.
<p><strong>Step 1:</strong> Apply De Morgan's Law to (A ∩ B')'</p><p>(A ∩ B')' = A' ∪ (B')' = A' ∪ B</p><p><strong>Step 2:</strong> Now the expression becomes (A' ∪ B) ∪ (B ∩ C)</p><p><strong>Step 3:</strong> Use associativity and commutativity: A' ∪ B ∪ (B ∩ C)</p><p><strong>Step 4:</strong> Since (B ∩ C) ⊆ B, we have B ∪ (B ∩ C) = B (absorption law)</p><p><strong>Step 5:</strong> Therefore: (A ∩ B')' ∪ (B ∩ C) = A' ∪ B</p><p>∴ Answer: B (which is A' ∪ B or equivalently B ∪ A')</p>
Correct Answer: B