Sets, Relations & Functions
Mathematical Reasoning
Grade 11
Question:
<p>If \((p \wedge \sim q) \wedge (p \wedge r) \rightarrow \sim p \vee q\) is false, then the truth values of <em>p</em>, <em>q</em> and <em>r</em> are, respectively,</p>
<p>F, T, F</p>
<p>T, F, T</p>
<p>T, T, T</p>
<p>F, F, F</p>
Step-by-Step Solution
Key Concept: A conditional statement A → B is false only when A is true and B is false. Use this to determine when (p ∧ ¬q) ∧ (p ∧ r) is true and ¬p ∨ q is false simultaneously.
<p><strong>Step 1:</strong> For the conditional (p ∧ ¬q) ∧ (p ∧ r) → ¬p ∨ q to be FALSE, we need:</p><ul><li>Antecedent TRUE: (p ∧ ¬q) ∧ (p ∧ r) = TRUE</li><li>Consequent FALSE: ¬p ∨ q = FALSE</li></ul><p><strong>Step 2:</strong> From ¬p ∨ q = FALSE:</p><ul><li>Both ¬p and q must be FALSE</li><li>Therefore: p = TRUE and q = FALSE</li></ul><p><strong>Step 3:</strong> From (p ∧ ¬q) ∧ (p ∧ r) = TRUE with p = TRUE and q = FALSE:</p><ul><li>(TRUE ∧ TRUE) ∧ (TRUE ∧ r) = TRUE</li><li>TRUE ∧ (TRUE ∧ r) = TRUE</li><li>TRUE ∧ r = TRUE</li><li>Therefore: r = TRUE</li></ul><p><strong>Step 4:</strong> Verify: (T ∧ T) ∧ (T ∧ T) → (F ∨ F) = T → F = FALSE ✓</p><p>∴ Answer: <strong>p = T, q = F, r = T</strong> (Option B)</p>
Correct Answer: B