Sets, Relations & Functions
Mathematical Reasoning
Grade 11
Question:
<p>If \(p\), \(q\) and \(r\) are simple propositions such that \((p \wedge q) \wedge (q \wedge r)\) is true, then</p>
<p>\(p\), \(q\) and \(r\) are all false</p>
<p>\(p\), \(q\) and \(r\) are all true</p>
<p>\(p\), \(q\) are true and \(r\) is false</p>
<p>\(p\) is true and \(q\), \(r\) are false</p>
Step-by-Step Solution
Key Concept: For a conjunction (AND) of multiple statements to be true, ALL individual statements must be true simultaneously. Therefore, if (p∧q)∧(q∧r) is true, then p, q, and r must each be true individually.
<p><strong>Step 1:</strong> Analyze the given compound proposition: (p∧q)∧(q∧r) is true</p><p><strong>Step 2:</strong> For a conjunction A∧B to be true, both A and B must be true. Here A = (p∧q) and B = (q∧r)</p><p><strong>Step 3:</strong> Since (p∧q) is true: both p is true AND q is true</p><p><strong>Step 4:</strong> Since (q∧r) is true: both q is true AND r is true</p><p><strong>Step 5:</strong> Combining all conditions: p ≡ T, q ≡ T, and r ≡ T</p><p><strong>Step 6:</strong> The compound statement simplifies to: (T∧T)∧(T∧T) = T∧T = T ✓</p><p><strong>Conclusion:</strong> All three propositions p, q, and r must be true.</p><p>∴ Answer: B</p>
Correct Answer: B