Sets, Relations & Functions
Mathematical Reasoning
Grade 11
Question:
<p>If \(p\), \(q\) and \(r\) are simple propositions with truth values T, F and T, respectively, then the truth value of \((\sim p \vee q) \wedge \sim r \rightarrow p\) is</p>
<p>True</p>
<p>False</p>
<p>True if \(r\) is false</p>
<p>True if \(q\) is true</p>
Step-by-Step Solution
Key Concept: Evaluate the compound proposition step-by-step using the given truth values (p=T, q=F, r=T), remembering that an implication A→B is false only when A is true and B is false.
<p><strong>Step 1:</strong> Identify the given truth values: p = T, q = F, r = T</p><p><strong>Step 2:</strong> Evaluate the components of (∼p ∨ q) ∧ ∼r → p</p><p>∼p = ∼T = F</p><p>∼r = ∼T = F</p><p><strong>Step 3:</strong> Evaluate (∼p ∨ q)</p><p>(F ∨ F) = F</p><p><strong>Step 4:</strong> Evaluate (∼p ∨ q) ∧ ∼r</p><p>F ∧ F = F</p><p><strong>Step 5:</strong> Evaluate the implication (∼p ∨ q) ∧ ∼r → p</p><p>F → T = T</p><p>(An implication is true when the antecedent is false, regardless of the consequent)</p><p><strong>∴ Answer: T (True)</strong></p>
Correct Answer: A