Sets, Relations & Functions
Mathematical Logic
Grade 11
Question:
<p>Given \((p \land {\sim}q) \land (p \land r) \to {\sim}p \lor q\) is false. Now with the given truth table, what is the only possible solution of \((p, q, r)\)?</p>
<p>(1) (T, F, T) or (T, F, F)</p>
<p>(2) (T, F, T) or (T, T, F)</p>
<p>(3) (F, T, T) or (F, F, F)</p>
<p>(4) (T, T, T) or (T, F, F)</p>
Step-by-Step Solution
Key Concept: An implication A → B is false only when A is true and B is false. Therefore, we need (p ∧ ¬q) ∧ (p ∧ r) to be true and ¬p ∨ q to be false simultaneously.
<p><strong>Step 1:</strong> For the implication to be false: Antecedent (p ∧ ¬q) ∧ (p ∧ r) must be TRUE and Consequent ¬p ∨ q must be FALSE.</p><p><strong>Step 2:</strong> For ¬p ∨ q to be false: ¬p = F and q = F, which gives p = T and q = F.</p><p><strong>Step 3:</strong> Substitute p = T and q = F into the antecedent: (T ∧ ¬F) ∧ (T ∧ r) = (T ∧ T) ∧ (T ∧ r) = T ∧ (T ∧ r)</p><p><strong>Step 4:</strong> For this to be true: T ∧ (T ∧ r) must be true, which requires r = T.</p><p><strong>Step 5:</strong> Verification: With (p, q, r) = (T, F, T):<br/>Antecedent: (T ∧ T) ∧ (T ∧ T) = T ✓<br/>Consequent: F ∨ F = F ✓<br/>Therefore T → F = F ✓</p><p>∴ Answer: (p, q, r) = <strong>(T, F, T)</strong> or <strong>B</strong></p>
Correct Answer: B