Sets, Relations & Functions
Mathematical Reasoning / Logic
Grade 11
Question:
<p>Given \(((p \wedge q) \vee (p \vee \sim q)) \wedge (\sim p \wedge \sim q)\) is equivalent to:</p>
<p>\(p \wedge (\sim q)\)</p>
<p>\((\sim p) \wedge (\sim q)\)</p>
<p>\(\sim (p \vee q)\) i.e. \(\sim p \wedge \sim q\)</p>
<p>\(p \vee q\)</p>
Step-by-Step Solution
Key Concept: Simplify the logical expression using Boolean algebra: first reduce the disjunction in the first bracket, then apply conjunction with the negation. The key is recognizing that (p ∧ q) ∨ (p ∨ ¬q) simplifies to p, making the entire expression reduce to a contradiction.
<p><strong>Step 1: Simplify the first bracket (p ∧ q) ∨ (p ∨ ¬q)</strong></p><p>Using associativity and commutativity: (p ∧ q) ∨ p ∨ ¬q</p><p>By absorption law: p ∨ (p ∧ q) = p, so we get: p ∨ ¬q</p><p><strong>Step 2: Apply the second bracket</strong></p><p>The full expression becomes: (p ∨ ¬q) ∧ (¬p ∧ ¬q)</p><p><strong>Step 3: Distribute the conjunction</strong></p><p>(p ∨ ¬q) ∧ (¬p ∧ ¬q) = [(p ∨ ¬q) ∧ ¬p] ∧ ¬q</p><p><strong>Step 4: Simplify (p ∨ ¬q) ∧ ¬p</strong></p><p>By distributivity: (p ∧ ¬p) ∨ (¬q ∧ ¬p) = F ∨ (¬p ∧ ¬q) = ¬p ∧ ¬q</p><p><strong>Step 5: Final result</strong></p><p>(¬p ∧ ¬q) ∧ ¬q = ¬p ∧ ¬q</p><p>∴ The expression is equivalent to <strong>¬p ∧ ¬q (or ¬(p ∨ q))</strong></p>
Correct Answer: C