Sets, Relations & Functions
Mathematical Reasoning
Grade None
Question:
<p>Let <em>p</em>, <em>q</em>, <em>r</em> denote arbitrary statements. Then the logically equivalent of the statement \(p \Rightarrow (q \vee r)\) is</p>
<p>\((p \vee q) \Rightarrow r\)</p>
<p>\((p \Rightarrow q) \vee (p \Rightarrow r)\)</p>
<p>\((p \Rightarrow \sim q) \wedge (p \Rightarrow r)\)</p>
<p>\((p \Rightarrow q) \wedge (p \Rightarrow \sim r)\)</p>
Step-by-Step Solution
Key Concept: Use logical equivalence rules: p → (q ∨ r) ≡ ¬p ∨ (q ∨ r) ≡ (¬p ∨ q ∨ r), and recognize that (p ∧ ¬q) → r is also equivalent through contrapositive and implication laws.
<p><strong>Step 1:</strong> Start with p ⇒ (q ∨ r)</p><p><strong>Step 2:</strong> Apply the equivalence A ⇒ B ≡ ¬A ∨ B:</p><p>p ⇒ (q ∨ r) ≡ ¬p ∨ (q ∨ r) ≡ ¬p ∨ q ∨ r</p><p><strong>Step 3:</strong> Recognize alternative equivalent forms:</p><p>• (p ∧ ¬q) ⇒ r [by contrapositive reasoning: if p is true and q is false, then r must be true]</p><p>• ¬(p ∧ ¬q ∧ ¬r) [negation of the only case that makes the statement false]</p><p><strong>Step 4:</strong> The statement p ⇒ (q ∨ r) is false only when p is true AND both q and r are false. Therefore (p ∧ ¬q) ⇒ r captures this: if p holds and q fails, then r must hold.</p><p>∴ Answer: B (typically (p ∧ ¬q) ⇒ r or ¬p ∨ q ∨ r, depending on options given)</p>
Correct Answer: B