# Proving the contrapositive

## Main Question or Discussion Point

It can be proved by proof by contradiction. hence it is just a variant of it?

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Not really. Say you have a statement p, a proof by contradiction would be not p implies false. Contrapositive is a direct proof. Say you have the statement p imples q. To prove this via contradiction you assume p is true and q is false then derive a contradiction, to prove this via contrapositive you assume not q is true (i.e. q is false) and prove that this imples p is false.

CRGreathouse
Homework Helper
Statement: p ⇒ q
Proof by contradiction: Assume p and ¬q; prove ⊥.
Proof by contraposition: Assume ¬q; prove ¬p.

A proof by contraposition can be turned into a proof by contradiction by adding the step "p ∧ ¬p ⇒ ⊥". But proofs by contradiction don't need to prove ¬p as an intermediate step; they can prove contradiction in other ways. So proof by contradiction is stronger.

On the other hand, the more constrained proof by contraposition is considered a constructive technique, while proof by contradiction is always considered nonconstructive. So if you have a choice use contraposition.

HallsofIvy