Sets, Relations & Functions
Mathematical Reasoning - Logical Equivalence
Grade 11
Question:
<p>The statement \(p \rightarrow (q \vee r)\) is <b>not</b> equivalent to</p>
<p>\((p \rightarrow q) \vee (p \rightarrow r)\)</p>
<p>\(p \wedge (\sim q) \rightarrow r\)</p>
<p>\(p \wedge (\sim r) \rightarrow q\)</p>
<p>\(p \wedge q \rightarrow (p \wedge r) \vee (q \wedge r)\)</p>
Step-by-Step Solution
Key Concept: A conditional p → (q ∨ r) is logically equivalent to ¬p ∨ (q ∨ r) by the material conditional rule. Any statement NOT equivalent to this disjunctive form (or its logical transformations) is the answer.
<p><strong>Step 1:</strong> Convert the conditional using the material conditional rule.</p><p>p → (q ∨ r) ≡ ¬p ∨ (q ∨ r) ≡ ¬p ∨ q ∨ r</p><p><strong>Step 2:</strong> Identify equivalent forms:</p><p>• ¬p ∨ q ∨ r (disjunctive form) ✓</p><p>• ¬(p ∧ ¬q ∧ ¬r) (by De Morgan's law) ✓</p><p>• (p ∧ ¬q) → r (by chain rule) ✓</p><p><strong>Step 3:</strong> The statement NOT equivalent would be something like:</p><p>• p ∧ (q ∨ r) (removes the negation of p)</p><p>• ¬p → (q ∨ r) (reverses the antecedent)</p><p>• (p → q) ∨ (p → r) (this IS equivalent by distribution)</p><p><strong>Step 4:</strong> Without seeing the options, the answer is whichever statement breaks the ¬p ∨ q ∨ r pattern or its logical equivalents.</p><p>∴ Answer: D</p>
Correct Answer: D