Question

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 :

• A. $\sim Q\leftrightarrow \sim P\vee R$
• B. $\sim Q\leftrightarrow P\wedge R$
• C. $\sim Q\leftrightarrow P\wedge \sim R$
• D. $\sim Q\leftrightarrow \sim P\wedge R$

Answer 1

Answer: (C)

$\sim Q\leftrightarrow P\wedge \sim R$

Given statement is "Suman is brilliant and dishonest if and only if Suman is rich"
$\Rightarrow (P\wedge \sim R)\leftrightarrow Q$
We know that, $\sim (a\leftrightarrow b)$
= $\sim b\leftrightarrow a$

Thus negation of given statement can be written as -
$\sim \left(\right(P\wedge \sim R)\leftrightarrow Q)$

$\sim Q\leftrightarrow (P\wedge \sim R)$

Hence, option C is correct.