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Clearly, there are standard ways of proving general theorems without regard to the actual subject matter. These include, but are not limited to, proof by contrapositive, proof by contradiction, and direct proof.

However, having taking an advanced course in Real Analysis, I noticed that there are many recurring proof techniques specific to the subject matter. The most obvious, perhaps, is the standard $\epsilon,\delta$ proof. Another technique, often used in dealing with families of functions, are $\frac{\epsilon}{3}$ arguments. Of course, there are others, but it seems as if much of the intuition in analysis pertains to understanding these and other proof techniques.

Now, I am in an Abstract Algebra course, and I have yet to fully grasp the proof techniques or the intuition. So, I have two questions. One, what are standard proof techniques in Algebra (or is there a compendium of sorts for such proof techniques)? Two, how does this generalize to other fields? It seems as if one understand the motivation behind the proof technique, then the intuition will be easier. As such, if there is such material, I would love to delve into it prior to me actually encountering material in the course which requires it.

Many thanks.

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I disagree strongly with the suggestion that "there are standard ways of proving general theorems without regard to the actual subject matter". –  Chris Godsil Feb 6 '13 at 2:41
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@ChrisGodsil Of course these ways are not unique, and not every proof must fall within these bounds. However, it is patently clear that a "proof by contradiction" is a standard way of proving certain theorems in all subjects. –  Jebruho Feb 6 '13 at 2:42
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But most (not all) proofs by contradiction can be written without using contradiction. I find that telling someone to "prove it by contradiction" is rarely helpful. –  Chris Godsil Feb 6 '13 at 2:52
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In the early days of an abstract algebra class, the standard proof technique is, "apply the definitions". –  Gerry Myerson Feb 6 '13 at 3:17
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In beginning Algebra, many "proof questions" are really language lessons. If one understands the relevant definition, the actual verification is usually straightforward. –  André Nicolas Feb 6 '13 at 3:18
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I hope you're still interested in this question! There's some great material here: http://www.tricki.org/article/Algebra_front_page. The website seems to be a ghost town these days, which is a shame, but the content doesn't go bad!

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