Negation of the Boolean statement ( p ∨ q ) ⇒((∼ r ) ∨ p ) is equivalent to:

Question

Negation of the Boolean statement ( p ∨ q ) ⇒((∼ r ) ∨ p ) is equivalent to: (A) p ∧(∼ q ) ∧ r (B) (∼ p ) ∧(∼ q ) ∧ r (C) (∼ p) ∧ q ∧ r (D) p ∧ q ∧(∼ r )

Tardigrade Admin · Accepted Answer

P ∨ q ⇒(∼ r ∨ p ) ≡ ∼( p ∨ q ) ∨(∼ r ∨ p ) ≡(∼ p ∧ ∼ q ) ∨( p ∨ ∼ r ) ≡[∼ p ∨ p ) ∧(∼ q ∨ p )] ∨ ∼ r ≡[∼ q ∨ p ) ∨ ∼ r Its negation is ∼ p ∧ q ∧ r

Q. Negation of the Boolean statement $(p \lor q) \Rightarrow ((\sim r) \lor p)$ is equivalent to:

$p \land (\sim q) \land r$

$(\sim p) \land (\sim q) \land r$

$(\sim p) \land q \land r$

$p \land q \land (\sim r)$

$P \lor q \Rightarrow (\sim r \lor p)$
$\equiv\sim (p \lor q) \lor (\sim r \lor p)$
$\equiv (\sim p \land \sim q) \lor (p \lor \sim r)$
$\equiv [\sim p \lor p) \land (\sim q \lor p)] \lor \sim r$
$\equiv [\sim q \lor p) \lor \sim r$
Its negation is $\sim p \land q \land r$

Q. Negation of the Boolean statement (p∨q)⇒((∼r)∨p) is equivalent to:

p∧(∼q)∧r

(∼p)∧(∼q)∧r

(∼p)∧q∧r

p∧q∧(∼r)