Question

Given the conditional statement ~p -> q which statement is logically equivalent?​

Answer 1

Any implication is logically equivalent to its contrapositive. In other words,

¬p ⇒ q   ⇔   ¬q ⇒ p

(¬ means the same thing as ~, "not")

To prove this: recall that

p ⇒ q   ⇔   ¬p ∨ q

This is because p ⇒ q is true if p is false, or both p and q are true, i.e.

p ⇒ q   ⇔   ¬p ∨ (p ∧ q)

Disjunction (∨ or "or") distributes over conjunction (∧ or "and"), so that

p ⇒ q   ⇔   (¬p ∨ p) ∧ (¬p ∨ q)

but ¬p ∨ p is always true, or a tautology, so we're just left with ¬p ∨ q.

Then

¬p ⇒ q   ⇔   p ∨ q

…   ⇔   q ∨ p

…   ⇔   ¬(¬q) ∨ p

…   ⇔   ¬q ⇒ p