# Substitute Proof by Contradiction/Negation with Direct Proof or Proof by Contrapositive

When do proof by contradiction, or by negation, or direct proof, or proof by contrapositive all work equally well? When can any one be chosen as the proof form?
I'm interested in a pragmatic answer and not a relatively theoretical one.

As mooted here, I always want to supplant any proof that's not a direct proof or proof by either such, thus I must firstly be enlightened that this switch is admissible.
Nevertheless, if a reference fails to advise/divulge that any one can be superseded,
how would a student divine/previse of this freedom of choice/interchange, before trying to prove?

In the following instances, I didn't know if direct proofs or by contrapositive had existed. If I hadn't Googled, I would've been benighted about easier proofs, a tribulation which I want to prevent.
In Linear Algebra 3rd ed, David Poole proves by contradiction P308 Theorem 4.20 (cp Theorem 1 in this PDF) and P316 Theorem 4.2.4, but doesn't impart the existence and validity of direct proofs.

P3 proves by contradiction the Exchange Theorem, yet it doesn't uncloak the direct proof:
P41 of Linear Algebra by Alain Robert.

I reference related questions: When to use the contrapositive to prove a statment, Proof by contradiction vs Prove the contrapositive.

• It is highly unlikely that just by the form of the question you should be able to conclude which proof technique will work, or which one can be substituted successfully for another. Besides, it is not particularly interesting to know when a given result can be proved in more than one way. The question of constructively valid proofs is more interesting. – Ittay Weiss Feb 17 '14 at 8:29
• what do you mean with WLOG?, also one link " not a relatively theoretical one".is broken so i don't really understand the question, it is a very theoretical one (isn't it?) (see also my quite theoretical answer at math.stackexchange.com/questions/638756/…. – Willemien Feb 17 '14 at 8:46
• Unless you are a specifically interested in this topic (e.g. just trying to better understand linear algebra), the distinctions and subtleties between types of proof are not worth the time, take the best tool available to you and just use it. On the other hand, if you are, you might be interested in mathematical constructivism, which is a whole branch that strives to use only constructive proofs (and consequences of disregarding other methods). – dtldarek Feb 17 '14 at 9:04
• @Willemien: I beg your pardon. I emended the link. I've also tried to elucidate my first paragraph and thus removed the WLOG. Better? – Greek - Area 51 Proposal Feb 18 '14 at 8:34
• This question gets more and more interesting, I came to the conclusion that indirect proof can always be replaced by an Indirect conditional proof and an Indirect Contrapositive proof by an Contrapositive Indirect proof and that the latter ones are preferable, will amend my anwser at the other question. – Willemien Feb 18 '14 at 12:22