Respuesta :

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

¬pq   ⇔   ¬qp

(¬ means the same thing as ~, "not")

To prove this: recall that

pq   ⇔   ¬pq

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

pq   ⇔   ¬p ∨ (pq)

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

pq   ⇔   (¬pp) ∧ (¬pq)

but ¬pp is always true, or a tautology, so we're just left with ¬pq.

Then

¬pq   ⇔   pq

…   ⇔   qp

…   ⇔   ¬(¬q) ∨ p

…   ⇔   ¬qp