negation of the statement "Suman is brilliant and dishonest if and only if Suman is rich"

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

(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?

Recall that $A\land B$ is logically equivalent to $\neg(A\to\neg B)$. — , Mar 23 '20 at 03:48
I really can not get your hint..Can you please write an answer?@user170039 — anonymous, Mar 23 '20 at 06:43

score 1 · Answer 1 · answered Mar 23 '20 at 04:03

Suman is brilliant and dishonest if and only if Suman is rich

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.

negation of the statement "Suman is brilliant and dishonest if and only if Suman is rich"

1 Answers1