Probability
Logic and Mathematical Reasoning
Grade 12
Question:
<p>Which of the following is a tautology?</p>
<p>(a) \((\sim p) \land (p \lor q) \to q\)</p>
<p>(b) \((q \to p) \lor \sim(p \to q)\)</p>
<p>(c) \((\sim q) \lor (p \land q) \to q\)</p>
<p>(d) \((p \to q) \land (q \to p)\)</p>
Step-by-Step Solution
Key Concept: A tautology is a compound statement that is always true regardless of the truth values of its components. Evaluate each option using a truth table to identify which one has all T values.
<p><strong>Step 1:</strong> Construct a truth table for option (a): \((\sim p) \land (p \lor q) \to q\)</p><p><strong>Step 2:</strong> Evaluate the compound statement for all possible truth values of p and q.</p><p><strong>Step 3:</strong> When p = T, q = T: \((\sim T) \land (T \lor T) \to T = F \land T \to T = F \to T = T\)</p><p><strong>Step 4:</strong> When p = T, q = F: \((\sim T) \land (T \lor F) \to F = F \land T \to F = F \to F = T\)</p><p><strong>Step 5:</strong> When p = F, q = T: \((\sim F) \land (F \lor T) \to T = T \land T \to T = T \to T = T\)</p><p><strong>Step 6:</strong> When p = F, q = F: \((\sim F) \land (F \lor F) \to F = T \land F \to F = F \to F = T\)</p><p>All truth values are T, therefore option (a) is a tautology. ∴ Answer is (a).</p>
Correct Answer: a