Let p, q, r be three statements. Then ∼ (p ∨ (q ∧ r)) is equal to

Question

Let p, q, r be three statements. Then ∼ (p ∨ (q ∧ r)) is equal to (A) (∼ p ∧ ∼ q)∧(∼ p ∧ ∼ r) (B) (∼ p ∨ ∼ q) ∧ (∼ p ∨ ∼ r) (C) (∼ p Lambda ∼ q) ∨ (∼ p Lambda ∼ r) (D) (∼ p ∨ ∼ q) ∨ (∼ p Lambda ∼ r)

Tardigrade Admin · Accepted Answer

∼ (p ∨(q ∧ r))=∼ p ∧ ∼(q ∧ r) =∼ p ∧(∼ q ∨ ∼ r) =(∼ p ∧ ∼ q) ∨(∼ p ∧ ∼ r)

Q. Let $p, q, r$ be three statements. Then $\sim (p \lor (q \land r))$ is equal to

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

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

$(\sim p Λ \sim q) \lor (\sim p Λ \sim r)$

$(\sim p \lor \sim q) \lor (\sim p Λ \sim r)$

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

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

Q. Let p,q,r be three statements. Then ∼(p∨(q∧r)) is equal to

(∼p∧∼q)∧(∼p∧∼r)

(∼p∨∼q)∧(∼p∨∼r)

(∼pΛ∼q)∨(∼pΛ∼r)

(∼p∨∼q)∨(∼pΛ∼r)

(∼p∧∼q)∨(∼p∨∼r)

∼(p∨(q∧r))=∼p∧∼(q∧r) =∼p∧(∼q∨∼r) =(∼p∧∼q)∨(∼p∧∼r)