Question

Consider the following statements:
P: Ramu is intelligent.
Q: Ramu is rich.
R: Ramu is not honest.
The negation of the statement "Ramu is intelligent and honest if and only if Ramu is not rich" can be expressed as:

• A. $\left(\right(P\wedge (\sim R))\wedge Q)\wedge \left(\right(\sim Q)\wedge ((\sim P)\vee R\left)\right)$
• B. $\left(\right(P\wedge R)\wedge Q)\vee \left(\right(\sim Q)\wedge ((\sim P)\vee (\sim R)\left)\right)$
• C. $\left(\right(P\wedge R)\wedge Q)\wedge \left(\right(\sim Q)\wedge ((\sim P)\vee (\sim R)\left)\right)$
• D. $\left(\right(P\wedge (\sim R))\wedge Q)\vee \left(\right(\sim Q)\wedge ((\sim P)\vee R\left)\right)$

Answer 1

The correct option is D $\left(\right(P\wedge (\sim R))\wedge Q)\vee \left(\right(\sim Q)\wedge ((\sim P)\vee R\left)\right)$

P: Ramu is intelligent.
Q: Ramu is rich.
R: Ramu is not honest.
Given statement, "Ramu is intelligent and honest if and only if Ramu is not rich"
$=(P\wedge \sim R\iff \sim Q$
So, negation of the statement is
$\sim \left[\right(P\wedge \sim R)\iff \sim Q]$
$=\sim \left[\right\{\sim (P\wedge \sim R)\vee \sim Q\}\wedge \{Q\vee (P\wedge \sim R)\left\}\right]$
$=\left(\right(P\wedge (\sim R))\wedge Q)\vee \left(\right(\sim Q)\wedge ((\sim P)\vee R\left)\right)$