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
(1) $\sim Q\leftrightarrow \sim P \wedge R$
(2) $\sim Q\leftrightarrow \sim P \vee R$
(3) $\sim Q\leftrightarrow P \vee\sim R$
(4) $\sim Q\leftrightarrow P \wedge \sim R$
My Attempt: I know that $\sim (A \leftrightarrow B) $ is equivalent to $( A \wedge \sim B) \vee ( B \wedge \sim A)$. BUt it does not equate to any of the option. And It will be very lengthy if I go to find the truth table of all the options. Can anyone please help me how to handle this ?
I was trying to solve this by using Truth Table also. Can anyone please check if it is correct or wrong?
This statement can be expressed as $Q \leftrightarrow (P\cdot\overline R )$,
Negating this with the help of this question, we get $\overline Q \leftrightarrow (P\cdot\overline R )$ or $Q \leftrightarrow \overline{(P\cdot\overline R )} = Q \leftrightarrow (\overline P + R)$
The first expression above corresponds to option (4), which is the answer.