Mathematical Reasoning
Proposition Logic and Truth Tables
Grade 11
Question:
<p>The statement <span>\((p \rightarrow (q \rightarrow p)) \rightarrow (p \rightarrow (p \vee q))\)</span> is</p>
<p>(a) equivalent to <span>\((p \wedge q) \vee (\neg q)\)</span></p>
<p>(b) a contradiction</p>
<p>(c) equivalent to <span>\((p \vee q) \wedge (\neg p)\)</span></p>
<p>(d) a tautology</p>
Step-by-Step Solution
Key Concept: A tautology is a statement that is true for all possible truth value assignments. Construct a complete truth table to verify.
<p><strong>Solution:</strong> Construct the truth table:</p><table style="border-collapse: collapse; width: 100%;"><tr><td style="border: 1px solid black; padding: 5px;"><span>\(p\)</span></td><td style="border: 1px solid black; padding: 5px;"><span>\(q\)</span></td><td style="border: 1px solid black; padding: 5px;"><span>\(p \vee q\)</span></td><td style="border: 1px solid black; padding: 5px;"><span>\(q \rightarrow p\)</span></td><td style="border: 1px solid black; padding: 5px;"><span>\(p \rightarrow (q \rightarrow p)\)</span></td><td style="border: 1px solid black; padding: 5px;"><span>\(p \rightarrow (p \vee q)\)</span></td><td style="border: 1px solid black; padding: 5px;"><span>\((p \rightarrow (q \rightarrow p)) \rightarrow (p \rightarrow (p \vee q))\)</span></td></tr><tr><td style="border: 1px solid black; padding: 5px;">T</td><td style="border: 1px solid black; padding: 5px;">T</td><td style="border: 1px solid black; padding: 5px;">T</td><td style="border: 1px solid black; padding: 5px;">T</td><td style="border: 1px solid black; padding: 5px;">T</td><td style="border: 1px solid black; padding: 5px;">T</td><td style="border: 1px solid black; padding: 5px;">T</td></tr><tr><td style="border: 1px solid black; padding: 5px;">T</td><td style="border: 1px solid black; padding: 5px;">F</td><td style="border: 1px solid black; padding: 5px;">T</td><td style="border: 1px solid black; padding: 5px;">T</td><td style="border: 1px solid black; padding: 5px;">T</td><td style="border: 1px solid black; padding: 5px;">T</td><td style="border: 1px solid black; padding: 5px;">T</td></tr><tr><td style="border: 1px solid black; padding: 5px;">F</td><td style="border: 1px solid black; padding: 5px;">T</td><td style="border: 1px solid black; padding: 5px;">T</td><td style="border: 1px solid black; padding: 5px;">F</td><td style="border: 1px solid black; padding: 5px;">T</td><td style="border: 1px solid black; padding: 5px;">T</td><td style="border: 1px solid black; padding: 5px;">T</td></tr><tr><td style="border: 1px solid black; padding: 5px;">F</td><td style="border: 1px solid black; padding: 5px;">F</td><td style="border: 1px solid black; padding: 5px;">F</td><td style="border: 1px solid black; padding: 5px;">T</td><td style="border: 1px solid black; padding: 5px;">T</td><td style="border: 1px solid black; padding: 5px;">T</td><td style="border: 1px solid black; padding: 5px;">T</td></tr></table><p>Since all values in the final column are T, the statement is a tautology.</p><p>Hence, option (d) is correct.</p>
Correct Answer: D