Sets, Relations & Functions
Mathematical Reasoning / Equivalence
Grade 11
Question:
<p><strong>Statement-1:</strong> \(\sim(p \leftrightarrow \sim q)\) is equivalent to \(p \leftrightarrow q\).<br><strong>Statement-2:</strong> \(\sim(p \leftrightarrow \sim q)\) is a tautology.</p>
<p>Statement-1 is true, Statement-2 is false.</p>
<p>Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.</p>
<p>Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.</p>
<p>Statement-1 is false, Statement-2 is true.</p>
Step-by-Step Solution
Key Concept: We need to evaluate the truth value of ~(p ↔ ~q) and compare it with p ↔ q using truth tables or logical equivalences. A biconditional is false when its two sides have different truth values.
<p><strong>Step 1: Analyze Statement-1 - Is ~(p ↔ ~q) equivalent to p ↔ q?</strong></p><p>Construct a truth table for both expressions:</p><table border='1' cellpadding='5'><tr><th>p</th><th>q</th><th>~q</th><th>p ↔ ~q</th><th>~(p ↔ ~q)</th><th>p ↔ q</th></tr><tr><td>T</td><td>T</td><td>F</td><td>F</td><td>T</td><td>T</td></tr><tr><td>T</td><td>F</td><td>T</td><td>T</td><td>F</td><td>F</td></tr><tr><td>F</td><td>T</td><td>F</td><td>T</td><td>F</td><td>F</td></tr><tr><td>F</td><td>F</td><td>T</td><td>F</td><td>T</td><td>T</td></tr></table><p>The columns for ~(p ↔ ~q) and p ↔ q are identical. Therefore, <strong>Statement-1 is TRUE</strong>.</p><p><strong>Step 2: Analyze Statement-2 - Is ~(p ↔ ~q) a tautology?</strong></p><p>From the truth table above, the column for ~(p ↔ ~q) shows: T, F, F, T</p><p>Since the expression is not true in all cases (it's false when p=T,q=F and when p=F,q=T), <strong>Statement-2 is FALSE</strong>. It is a contingency, not a tautology.</p><p><strong>Step 3: Determine the correct option.</strong></p><p>Statement-1 is true and Statement-2 is false.</p><p><strong>∴ Answer: A</strong></p>
Correct Answer: A