Sets, Relations & Functions
Mathematical Reasoning
Grade 11
Question:
<p>If \(p \rightarrow (q \vee r)\) is false, then the truth values of \(p\), \(q\) and \(r\) are, respectively,</p>
<p>T, T, F</p>
<p>F, F, F</p>
<p>F, T, T</p>
<p>T, F, F</p>
Step-by-Step Solution
Key Concept: A conditional p → q is false if and only if p is true and q is false. Here, p → (q ∨ r) is false means p must be true AND (q ∨ r) must be false, which requires both q and r to be false.
<p><strong>Step 1:</strong> Recall that a conditional statement p → q is false if and only if the antecedent p is <strong>true</strong> and the consequent q is <strong>false</strong>.</p><p><strong>Step 2:</strong> Apply this to p → (q ∨ r): For this to be false, we need:<br>• p = <strong>True</strong><br>• (q ∨ r) = <strong>False</strong></p><p><strong>Step 3:</strong> For (q ∨ r) to be false, both q and r must be false (since a disjunction is true if at least one disjunct is true):<br>• q = <strong>False</strong><br>• r = <strong>False</strong></p><p><strong>Step 4:</strong> Verify: p → (q ∨ r) = T → (F ∨ F) = T → F = <strong>False</strong> ✓</p><p>∴ Answer: p = <strong>True</strong>, q = <strong>False</strong>, r = <strong>False</strong> (respectively: <strong>T, F, F</strong>)</p>
Correct Answer: D