The statement pattern (∼ p ∧ q) is logically equivalent to

Question

The statement pattern (∼ p ∧ q) is logically equivalent to (A) (p ∨ q) ∨ ∼ p  (B) (p ∨ q) ∧ ∼ p  (C) (p ∧ q) → p  (D) (p ∨ q) → p

Tardigrade Admin · Accepted Answer

B arrow( p ∨ q ) ∧ ∼ p =( p ∧ ∼ p ) ∨( q ∧ ∼ p ) ldots Distributive law = F ∨ q (∧ ∼ p ) ldots Complementary law = q ∧ ∼ p ldots Identify law =∼ p ∧ q ldots Commutative law Hence, ∼ p ∧ q is equivalent to ( p ∨ q ) ∧ ∼ p

Q. The statement pattern $(\sim p \land q)$ is logically equivalent to

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

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

$(p \land q) \to p$

$(p \lor q) \to p$

$B \to (p \lor q) \land \sim p = (p \land \sim p) \lor (q \land \sim p) \dots$ Distributive law
$= F \lor q (\land \sim p) \dots$ Complementary law
$= q \land \sim p \dots$ Identify law
$=\sim p \land q \dots$ Commutative law
Hence, $\sim p \land q$ is equivalent to $(p \lor q) \land \sim p$

Q. The statement pattern (∼p∧q) is logically equivalent to

(p∨q)∨∼p

(p∨q)∧∼p

(p∧q)→p

(p∨q)→p