Consider the following statements :
P : Suman is brilliant.
Q : Suman is rich.
R : Suman is honest.
The negation of the statement, "Suman is brilliant and dishonest if and only if Suman is rich" can be equivalently expressed as :
P : Suman is brilliant.
Q : Suman is rich.
R : Suman is honest.
The negation of the statement, "Suman is brilliant and dishonest if and only if Suman is rich" can be equivalently expressed as :
Show Hint
- \(\sim Q\leftrightarrow\ \sim P\wedge R\)
- \(\sim Q\leftrightarrow\ \sim P\vee R\)
- \(\sim Q\leftrightarrow P\vee\ \sim R\)
- \(\sim Q\leftrightarrow P\wedge\ \sim R\)
The Correct Option is D
Solution and Explanation
Step 2: \(\sim(A\leftrightarrow B)\equiv A\leftrightarrow\ \sim B\) (the biconditional is false exactly when the two sides differ).
Step 3: So the negation is \((P\wedge\sim R)\leftrightarrow\ \sim Q\), i.e. \(\sim Q\leftrightarrow P\wedge\ \sim R\).
Answer: (4) \(\sim Q\leftrightarrow P\wedge\ \sim R\)
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