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)?
- Anybody can ask a question
- Anybody can answer
- The best answers are voted up and rise to the top
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.
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.
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.