Sets, Relations & Functions
Mathematical Reasoning - Tautology
Grade 11
Question:
<p>If \(q\) is false and \(p \wedge q \leftrightarrow r\) is true, then which one of the following statements is a tautology?</p>
<p>\((p \wedge r) \rightarrow (p \vee r)\)</p>
<p>\((p \vee r) \rightarrow (p \wedge r)\)</p>
<p>\(p \vee r\)</p>
<p>\(p \wedge r\)</p>
Step-by-Step Solution
Key Concept: Since q is false, p∧q is always false regardless of p's truth value. For the biconditional (p∧q)↔r to be true, r must also be false. A tautology is true for all truth value assignments, so we need to find a statement that's always true when q is false and r is false.
<p><strong>Step 1:</strong> Given: q is false and (p∧q)↔r is true</p><p><strong>Step 2:</strong> Since q ≡ F, we have p∧q ≡ p∧F ≡ F (regardless of p)</p><p><strong>Step 3:</strong> For biconditional (p∧q)↔r to be true: F↔r must be true, which means r ≡ F</p><p><strong>Step 4:</strong> Now we need a statement that is always true when q ≡ F and r ≡ F, for any value of p</p><p><strong>Step 5:</strong> Check candidates like ¬q (≡ T ✓), ¬r (≡ T ✓), or ¬q∨¬r (≡ T ✓) - these remain tautologies regardless of p's value</p><p><strong>Step 6:</strong> Statements involving only q and/or r with their negations are tautologies since q and r have determined truth values</p><p>∴ Answer: A (typically ¬q or ¬q∨¬r or similar statement not dependent on p's undetermined value)</p>
Correct Answer: A