Sets, Relations & Functions
Mathematical Reasoning / Negation
Grade 11
Question:
<p>Consider the following statements:<br>\(P\): Suman is brilliant.<br>\(Q\): Suman is rich.<br>\(R\): Suman is honest.<br>The negation of the statement "Suman is brilliant and dishonest if and only if Suman is rich" can be expressed as</p>
<p>\(\sim Q \leftrightarrow (\sim P \wedge R)\)</p>
<p>\(\sim Q \leftrightarrow (\sim P \wedge R)\)</p>
<p>\(\sim(P \vee \sim R) \leftrightarrow Q\)</p>
<p>\(\sim P \vee (Q \leftrightarrow \sim R)\)</p>
Step-by-Step Solution
Key Concept: The negation of a biconditional statement (A ↔ B) is (A ∧ ¬B) ∨ (¬A ∧ B), which represents when the two sides have different truth values. First translate the English statement into logical form, then apply negation rules carefully.
<p><strong>Step 1:</strong> Translate the statement into logical form. 'Suman is brilliant and dishonest if and only if Suman is rich' means: (P ∧ ¬R) ↔ Q</p><p><strong>Step 2:</strong> Apply the negation rule for biconditional. The negation of (A ↔ B) is (A ∧ ¬B) ∨ (¬A ∧ B), which means A and B have opposite truth values.</p><p><strong>Step 3:</strong> Therefore: ¬[(P ∧ ¬R) ↔ Q] = [(P ∧ ¬R) ∧ ¬Q] ∨ [¬(P ∧ ¬R) ∧ Q]</p><p><strong>Step 4:</strong> Simplify the second part: ¬(P ∧ ¬R) = (¬P ∨ R), so the negation becomes: [(P ∧ ¬R ∧ ¬Q) ∨ (¬P ∨ R) ∧ Q]</p><p><strong>Step 5:</strong> This can be expressed as: 'Either Suman is brilliant and dishonest but not rich, or Suman is rich but either not brilliant or honest (or both)'</p><p>∴ Answer: A</p>
Correct Answer: A