Sets, Relations & Functions
Mathematical Logic - Negation
Grade 11
Question:
<p>Consider the following statements<br>
<em>P</em> : Suman is brilliant<br>
<em>Q</em> : Suman is rich<br>
<em>R</em> : Suman is honest<br>
The negation of the statement "Suman is brilliant and dishonest if any only if Suman is rich" can be expressed as</p>
<p>\(\sim(P \wedge \sim R) \leftrightarrow Q\)</p>
<p>\(\sim P \wedge (Q \leftrightarrow \sim R)\)</p>
<p>\(\sim(Q \leftrightarrow (P \wedge \sim R))\)</p>
<p>\(\sim Q \leftrightarrow \sim P \wedge R\)</p>
Step-by-Step Solution
Key Concept: Identify that 'P and not R iff Q' means (P ∧ ¬R) ↔ Q, then negate using ¬(A ↔ B) ≡ (A ∧ ¬B) ∨ (¬A ∧ B), which represents XOR.
<p><strong>Step 1:</strong> Translate the given statement.</p><p>'Suman is brilliant and dishonest if and only if Suman is rich' means: (P ∧ ¬R) ↔ Q</p><p><strong>Step 2:</strong> Apply negation to the biconditional.</p><p>¬[(P ∧ ¬R) ↔ Q] ≡ (P ∧ ¬R) XOR Q</p><p><strong>Step 3:</strong> Expand using the XOR definition.</p><p>(A ↔ B is false) means exactly one of A or B is true, so:<br/>¬[(P ∧ ¬R) ↔ Q] ≡ [(P ∧ ¬R) ∧ ¬Q] ∨ [¬(P ∧ ¬R) ∧ Q]</p><p><strong>Step 4:</strong> Simplify the second part.</p><p>¬(P ∧ ¬R) ≡ (¬P ∨ R)</p><p><strong>Step 5:</strong> Final form.</p><p>∴ Answer: [(P ∧ ¬R ∧ ¬Q) ∨ (¬P ∨ R) ∧ Q)] or equivalently (Suman is brilliant and dishonest and not rich) OR (Suman is not brilliant or honest, and rich)</p>
Correct Answer: C