<p>If the statements \((p \wedge \sim r) \rightarrow (q \vee r)\), \(q\) and \(r\) are all false, then \(p\)</p>
Step-by-Step Solution
Key Concept: When a conditional statement (p ∧ ∼r) → (q ∨ r) is true and both q and r are false, the antecedent (p ∧ ∼r) must be false (since a true conditional with false consequent requires false antecedent). With r false, ∼r is true, so p must be false to make the antecedent false.
<p><strong>Step 1:</strong> Given that (p ∧ ∼r) → (q ∨ r) is true, and q = F, r = F.</p><p><strong>Step 2:</strong> Evaluate the consequent: q ∨ r = F ∨ F = F</p><p><strong>Step 3:</strong> For the conditional A → B to be true when B is false, A must be false (since T → F = F, which contradicts our premise).</p><p><strong>Step 4:</strong> Therefore (p ∧ ∼r) must be false.</p><p><strong>Step 5:</strong> Since r = F, we have ∼r = T. So (p ∧ T) must be false, which means p must be false.</p><p>∴ Answer: p is <strong>false</strong> (Answer: B)</p>
Correct Answer: B