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As one studies mathematics, how important it is to find various proofs? I mean that if one has to prove for example that conditions (i), (ii), and (iii) are equivalent then is it enough to learn to prove that (i)->(ii)->(iii)->(i) or do mathematicians also think why (ii)->(i), (i)->(iii) and (iii)->(ii)?

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An example I remember being presented this way is WO->ZL->AC->WO... –  Dan Brumleve May 22 '12 at 7:51
    
Basically if you are asked to prove the equivalence of (i), (ii), (iii) and you want to do it using a chain of implications, you would probably prefer to choose the chain which gives you the easiest proof. So you will probably think about all implications and try to show some of them, hoping that in the and you will find a "path" between any two of them. Maybe you will finish with (i)->(ii)->(iii), maybe with (i)->(iii)->(ii), maybe with something completely different. (E.g. the suggestion from Brian's answer, or you might find 4th equivalent condition which makes your proof simpler.) –  Martin Sleziak May 22 '12 at 8:24
    
@Martin Sleziak: Yes. I was thinking the situation such that if I have never proved the implication (ii)->(i) and then after years some student asks me to show it then is it more useful to start thinking why (ii)->(i) or to show that (ii)->(iii)->(i), which might be somewhat clumsier way to prove it. –  student May 22 '12 at 8:33
    
@student You do not need to think about a particular implication just because a student might ask you after some years. A student might also ask you if you can prove that theorem without using a particular lemma or while standing on your head. Yes, possibly I do not immediately know the perfect bijection between the zillion objects counted by Catalan numbers, but there is a law of diminishing returns there, as long as you do know different proof methods and can prove the equivalence somehow. –  Phira May 22 '12 at 11:13
    
When I was a student, one of my professors repeatedly gave wrong proofs. Pointing out the errors in the course did not help, because the professor would then conveniently claim that the proof was trivial. I went to my TA for help and he had a very, very hard time proving it and the proofs were not enlightening because the problem is that you have to set up the whole course in a certain way to be able to give good proofs. There is some value asking yourself whether another order of proof is better, but for standard results from linear algebra, you can assume that someone already checked. –  Phira May 22 '12 at 11:16

3 Answers 3

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Sometimes a direct proof of $(ii)\Rightarrow(iii)$, say, is difficult, and it’s easier to prove $(ii)\Leftrightarrow(i)\Leftrightarrow(iii)$ than to prove the ring of implications $(i)\Rightarrow(ii)\Rightarrow(iii)\Rightarrow(i)$. In any case, thinking about the various pairwise implications is likely to give you more insight into the matter than limiting yourself to the bare minimum necessary to establish the result.

More generally, a result may have several very different proofs using very different techniques. In such cases the different proofs are very likely to cast the result in very different lights, thereby illuminating it better than any single proof could. One example is Szemerédi's theorem, which now has proofs using three completely different techniques stemming from three areas of mathematics that are on the face of it very different indeed.

I would say that in general learning the specific proofs is much less important than learning the techniques that they use, in hopes of being able to use them oneself in some other setting. And since the more tools you have, the better, familiarity with more than one approach to proving a theorem can indeed be well worthwhile.

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Certainly, (i)-->(ii)-->(iii)-->...-->(i) will suffice to show that all are equivalent, and if all you want from life is to show this batch to be equivalent, then you will be happy. However, to show that (i) is equivalent to (j) directly may shed more light on the situation. The proof technique above shows how (i) is related to (ii) but does not show a direct relationship between (i) and (iii). Rather the relationship shown is a bit indirect in that you are going through (ii) first. I guess the lesson is that if you show such equivalences in different ways, you will gain different insights in the ways in which the various statements relate. Unfortunately, I do not have an example on the top of my head at the moment, but if I think of one, I will provide it.

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Proving the chain of implications (i)->(ii)->(iii)->(i) is a standard technique used in textbooks, and works perfectly. Why? The trick here is that it loops, so you can extend it to (i)->(ii)->(iii)->(i)->(ii)->(iii)->(i)->(ii)->(iii)->(i), or however long you want.

Assume that one of the conditions is true. Then all conditions after in that chain must be true, since you have proven the implication. But since it loops, all of them have to be true.

Assume one of them is false. Then all conditions before it in the chain has to be false (by contrapositivity). But since the chain loops, all of the conditions must be false.

Hence, since they are all either false or true, they are equivalent.

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The best example I can think of is in Lay: Linear algebra, where he proves somewhere close to 20 conditions equivalent to a matrix being invertible, and he uses that technique a lot, and at least once with as many as five conditions. –  Arthur May 22 '12 at 7:58
    
I think the question is about the amount of insight provided by this technique, not its validity. –  Dan Brumleve May 22 '12 at 8:09

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