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One of my friends who got 100 % in logic told me, "when in doubt use contradiction".

When I do proofs I personally find using contradiction a lot easier. Also, from the fact that a lot of famous theorems are proved using contradiction could it be that proof by contradiction is easier or inherently more powerful than other methods of proof?

Are there any topics that discuss such things?

Mark
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    See related, possible duplicate: http://math.stackexchange.com/questions/240/are-the-proofs-by-contradiction-weaker-than-other-proofs – JDH Jul 11 '11 at 02:52
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    Also related: http://mathoverflow.net/questions/12342/reductio-ad-absurdum-or-the-contrapositive – Qiaochu Yuan Jul 11 '11 at 02:57
  • @Mark: So-called "direct" proofs often have a more constructive character. But the difference is often illusory. For example, the Cantor diagonalization argument, which is often cited as a proof by contradiction of the existence of transcendental numbers, can be turned in a mechanical way into an explicit construction of a transcendental number. – André Nicolas Jul 11 '11 at 03:19
  • @JDH I understand that in logic proof by contradiction and proof by construction are logically equivalent. But that doesn't explain by so many theorems use contradiction (e.g. is it possible to construct an irrational number? I am not sure). Or why some theorems become alot easier to prove when using contradiction. – Mark Jul 11 '11 at 03:19
  • @user6312 do you know of a theorem that must use contradiction to prove? – Mark Jul 11 '11 at 03:21
  • @Mark: A much discussed good candidate is the irrationality of $\sqrt{2}$. The reason is that the definition of irrational has the negative built in. We could find a positive characterization of some irrationals, for example by using continued fractions. But at least for that attempted bypass, the argument has a built in proof by contradiction at an earlier stage. – André Nicolas Jul 11 '11 at 03:32
  • @Mark: Your friend's remark is probably good exam-taking strategy. If we want to show that $P\implies Q$, and there is no obvious direct path, it can be useful to ask: How could $Q$ go wrong? – André Nicolas Jul 11 '11 at 03:36
  • One can view the usual proof of the irrationality of $\sqrt{2}$ as establishing the completely positive universal claim that every rational number has a square either strictly larger than $2$ or strictly less than $2$. – JDH Jul 11 '11 at 03:39
  • @Mark : Your question to my mind is to the effect that, 'Is any logical deduction using mathematics is trivial ?'...well one can believe any way one wants, but personally I feel that you lose fun when you try to go by blind thumb rules in mathematics, you can seek help from others but you cannot wish you ad a genie which solves any math problem. – Rajesh D Jul 11 '11 at 03:56
  • I'll note that a lot of automated theorem provers work by assuming the hypotheses and the negation of the conclusion, and attempt to derive a contradiction. This is probably because it's a somewhat less open-ended search. – Zhen Lin Jul 11 '11 at 04:21
  • @Rajesh are you implying something in particular? – Mark Jul 11 '11 at 05:07
  • @Mark : No but now that you have asked,If you really need help from this site, try asking questions/problems involving mathematics to a large extent. With my mediocre math brain, I have a strong feeling that you are not going to get anything out of questions like this. – Rajesh D Jul 11 '11 at 05:15
  • @Rajesh there is a reason why I tagged this proof-strategy. – Mark Jul 11 '11 at 05:19
  • @Rajesh you seem to be provoked by something I did not say. Anyways I am ending this conversation here since it's becoming non-constructive. – Mark Jul 11 '11 at 05:23
  • @Mark : Sorry for that. I was just having a bad Monday when i came across this question. Any way It was fun talking to you and hope you enjoyed this conversation. – Rajesh D Jul 11 '11 at 05:27
  • @Rajesh Don't worry about it I have bad days all the time. – Mark Jul 11 '11 at 05:37

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Finding a proof by contradiction often is easier, because you have more to work with. Suppose that you’re trying to prove an implication $\phi \to \psi$. For a direct proof you have only $\phi$ to work with (plus whatever related facts you may know). Similarly, for a proof of the contrapositive you have only $\lnot\psi$. For a proof by contradiction, however, you have both $\phi$ and $\lnot\psi$ at your disposal. Every direct consequence of $\phi$ is available to you, and so is every direct consequence of $\lnot\psi$.

Not infrequently you still end up with a direct proof of either $\phi \to \psi$ or its contrapositive, or something that can very easily be turned into such a proof, but this doesn’t negate the benefit of having more resources on hand when you start looking for a proof.

That said, I agree with user6312 that some results often presented (and perhaps most easily found) as proofs by contradiction would be more illuminating if presented in some other way. I’d even use the same example, though in a slightly different way: Cantor’s diagonal argument is really just a machine for producing a real number that isn’t in a given countable list of real numbers.

Brian M. Scott
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