Let p, q and r be three statements, then (∼ p arrow q) arrow r is equivalent to

Question

Let p, q and r be three statements, then (∼ p arrow q) arrow r is equivalent to (A) (∼ p ∨ r)∧ (q ∨ r) (B) (p arrow r)∧ (q arrow r) (C) ( text~ textitp ∧ r)∨ (q ∨ r) (D) (p arrow q) arrow r

Tardigrade Admin · Accepted Answer

(∼ p arrow q) arrow r≡ (p ∨ q) arrow r ≡ ∼ (p ∨ q)∨ r ≡ (∼ p ∧ ∼ q)∨ r ≡ (∼ p ∨ r)∧ (∼ q ∨ r)≡ (p arrow r)∧ (q arrow r)

Q. Let $p, q$ and $r$ be three statements, then $(\sim p \to q) \to r$ is equivalent to

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

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

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

$(p \to q) \to r$

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

Q. Let p,q and r be three statements, then (∼p→q)→r is equivalent to

(∼p∨r)∧(q∨r)

(p→r)∧(q→r)

( p∧r)∨(q∨r)

(p→q)→r