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 :

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) (( P ∧(∼ R )) ∧ Q ) ∧((∼ Q ) ∧((∼ P ) ∨ R )) (B) (( P ∧ R ) ∧ Q ) ∨((∼ Q ) ∧((∼ P ) ∨(∼ R ))) (C) (( P ∧ R ) ∧ Q ) ∧((∼ Q ) ∧((∼ P ) ∨(∼ R ))) (D) (( P ∧(∼ R )) ∧ Q ) ∨((∼ Q ) ∧((∼ P ) ∨ R ))

Tardigrade Admin · Accepted Answer

Negation of ( P ∧ ∼ R ) rightarrow(∼ Q ) ⇒(( P ∧ ∼ R ) ∧ Q ) ∨(∼ Q ∧ ∼( P ∧ ∼ R )) ⇒(( P ∧ ∼ R ) ∧ Q ) ∨(∼ Q ∧(∼ P ∨ R )) Answer D is correct

Q. 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 :

$((P \land (\sim R)) \land Q) \land ((\sim Q) \land ((\sim P) \lor R))$

$((P \land R) \land Q) \lor ((\sim Q) \land ((\sim P) \lor (\sim R)))$

$((P \land R) \land Q) \land ((\sim Q) \land ((\sim P) \lor (\sim R)))$

$((P \land (\sim R)) \land Q) \lor ((\sim Q) \land ((\sim P) \lor R))$

Negation of $(P \land \sim R) \leftrightarrow (\sim Q)$
$\Rightarrow ((P \land \sim R) \land Q) \lor (\sim Q \land \sim (P \land \sim R))$
$\Rightarrow ((P \land \sim R) \land Q) \lor (\sim Q \land (\sim P \lor R))$
Answer $D$ is correct

Q. 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 :

((P∧(∼R))∧Q)∧((∼Q)∧((∼P)∨R))

((P∧R)∧Q)∨((∼Q)∧((∼P)∨(∼R)))

((P∧R)∧Q)∧((∼Q)∧((∼P)∨(∼R)))

((P∧(∼R))∧Q)∨((∼Q)∧((∼P)∨R))