Sets, Relations & Functions
Mathematical Reasoning / Tautology
Grade 11
Question:
<p>Which one of the following Boolean expressions is a tautology?</p>
<p>\((p \wedge q) \vee (p \wedge \sim q)\)</p>
<p>\((p \vee q) \vee (p \vee \sim q)\)</p>
<p>\((p \vee q) \wedge (p \vee \sim q)\)</p>
<p>\((p \vee q) \wedge (\sim p \vee \sim q)\)</p>
Step-by-Step Solution
Key Concept: A tautology is a Boolean expression that is true for ALL possible values of its variables. Test each option by constructing truth tables or identifying when the expression could be false.
<p><strong>Step 1:</strong> Recall that a tautology has truth value TRUE for every possible assignment of truth values to its variables.</p><p><strong>Step 2:</strong> For each option, construct a truth table or identify a case where the expression could be false:</p><p>• Expressions with AND only: False when any variable is false</p><p>• Expressions with OR only: False when all variables are false</p><p>• Expressions with implications: False only when antecedent is TRUE and consequent is FALSE</p><p><strong>Step 3:</strong> The tautology must remain TRUE in all cases. The most common tautologies are:</p><p>• p ∨ ¬p (law of excluded middle)</p><p>• p → p</p><p>• (p ∧ q) → p</p><p>• ¬(p ∧ ¬p) (law of non-contradiction)</p><p><strong>Step 4:</strong> Option B satisfies the condition of being true for all possible truth assignments.</p><p>∴ Answer: B</p>
Correct Answer: B