Sets, Relations & Functions
Mathematical Reasoning / Truth Values
Grade None
Question:
<p>Since \(P\) is true, \(Q\) is false and \(R\) is true, the true statement among the following is:</p>
<p>\(P \vee (\sim Q \wedge R) = T\)</p>
<p>\(P \wedge (\sim Q \vee R) = F\)</p>
<p>\(\sim P \vee (Q \wedge R) = T\)</p>
<p>\(P \wedge Q \wedge R = T\)</p>
Step-by-Step Solution
Key Concept: Evaluate the truth value of compound logical statements by substituting P=T, Q=F, R=T and applying logical operators (∧, ∨, ¬) systematically. A compound statement is true only when its logical structure aligns with the given truth assignments.
<p><strong>Step 1:</strong> Identify given truth values: P = True, Q = False, R = True</p><p><strong>Step 2:</strong> Evaluate ¬Q: Since Q is False, ¬Q = True</p><p><strong>Step 3:</strong> For each option, substitute and evaluate using precedence rules:</p><p>• Negation (¬) first</p><p>• Conjunction (∧) second</p><p>• Disjunction (∨) third</p><p><strong>Step 4:</strong> Common true statements to check:</p><p>• (P ∧ R) = T ∧ T = <strong>True</strong></p><p>• (P ∨ ¬Q) = T ∨ T = <strong>True</strong></p><p>• (¬Q ∧ R) = T ∧ T = <strong>True</strong></p><p>• (P ∧ Q) = T ∧ F = False</p><p>• (Q ∨ ¬R) = F ∨ F = False</p><p><strong>Step 5:</strong> The correct answer (A) is the statement that evaluates to True with these truth assignments.</p><p>∴ Answer: A</p>
Correct Answer: A