<p>Which of the following is true regarding \(p \to (q \to p)\)?</p>
<p>(1) It is a contradiction</p>
<p>(2) It is equivalent to \(p \to (p \lor q)\)</p>
<p>(3) It is equivalent to \(p \to (p \land q)\)</p>
<p>(4) None of these</p>
Step-by-Step Solution
Key Concept: Recognize that p → (q → p) is a tautology because whenever p is true, the consequent (q → p) is automatically true regardless of q's value; and when p is false, the implication p → (q → p) is vacuously true.
<p><strong>Step 1:</strong> Construct the truth table for p → (q → p)</p><p><table border='1' cellpadding='8'><tr><th>p</th><th>q</th><th>q → p</th><th>p → (q → p)</th></tr><tr><td>T</td><td>T</td><td>T</td><td>T</td></tr><tr><td>T</td><td>F</td><td>T</td><td>T</td></tr><tr><td>F</td><td>T</td><td>F</td><td>T</td></tr><tr><td>F</td><td>F</td><td>T</td><td>T</td></tr></table></p><p><strong>Step 2:</strong> Analyze each row:<br>• Row 1: p=T, q=T → q→p is T, so T→T = T<br>• Row 2: p=T, q=F → q→p is T, so T→T = T<br>• Row 3: p=F, q=T → q→p is F, so F→F = T (implication with false antecedent)<br>• Row 4: p=F, q=F → q→p is T, so F→T = T</p><p><strong>Step 3:</strong> Since the final column is all T, the formula p → (q → p) is a <strong>tautology</strong> (always true).</p><p>∴ Answer: B (This is a tautology)</p>
Correct Answer: B