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I want to start out by saying that I am new at proof based mathematics. I am used to seeing patterns and using them to solve similar problems. However, I have found this is not a very good way to study for courses in abstract mathematics! I have noticed that similar themes in mathematics keep coming up over and over again(for example, homomorphisims of objects). And partially because of this, we also have recurring proof methods. Many times I initially struggle with proofs because I don't think, "What are the proof methods available to solve this sort of problem?".

I have found that stopping to think about proof strategies is the best way to solve a proof based problem, but many times, because of inexperience I lack knowledge about what tools are available to me in this regard. For example, to prove $A=B$, a way to attack this problem is to try to show that $A\leq B$, and also that $A\geq B$. This proof strategy came up today when I was trying to prove $G_{b}=gG_{a}g^{-1}$ where $G_{x}$ is the stabilizer of the element $x$ under appropriate conditions. Anyway, I showed that $G_{b}\subseteq gG_{a}g^{-1}$ and also that $gG_{a}g^{-1}\subseteq G_{b}$. Once I realized how to attack this problem, the problem sort of solved itself. I have noticed that many problems sort of tell you how to solve them.

What are your favorite proof strategies, or ones that you have found extremely useful?

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General strategies abound but don't necessarily help much, while more effective, specialized strategies will be very different from area to area in math. – anon Oct 18 '11 at 22:36
related – t.b. Oct 18 '11 at 22:36
The complete answer will depend on what sort of logic one takes for granted. Mathematics based on intuitionist logic, for example, has different proof techniques available than mathematics based on classical logic. – Doug Spoonwood Oct 18 '11 at 23:19
@anon, so is there a resource, maybe like a book you recommend to practice these strategies? I am particularly interested in becoming better at abstract algebra type problems. – Edison Oct 19 '11 at 20:45
Not really. I'm better at analysis and basic logic than anything, and really I could use such a resource when it comes to abstract algebra problems. (The best I can do is recommend Arturo...) – anon Oct 19 '11 at 22:29

There are many interesting problem-solving techniques listed at the Tricki.

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Here is a list of 36 methods of proof.

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Awesome, it made my day +1 ;) – Fabien Jun 20 '14 at 2:54

Maybe this book can help you a little:

Daniel J. Velleman, How to Prove It: A Structured Approach

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Tim Gowers is discussing issues related to this question in an on-going series of blog posts. In particular, he is discussing basic logic, unwrapping definitions to get to the heart of a problem, and so on.

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well, it's good you are new to mathematical proof. there are many ways that you can prove the same result but it would decide how much effort you have to put depending on your selection.

if you already have the result, Induction will be a good choice. There are two forms of induction by the way. strong form and weak form.

otherwise, you will need to show a result is wrong. This is the easiest to show. You can take a counter example and show that the given result doesn't stand.

e.g: show that x+1 <0 for all x>0;

you take x=1 and 1+1=2 >0 so the given result is wrong.

contradiction is another famous method. You are asked to prove something. Then you accept that the negation is correct and try to prove that. Then you show that what you thought was wrong. Then the given answer should be correct. This method is really good when you have only two possible cases.

Graphical proofs are also a popular method in some cases but they don't give a rigorous proof in most of the times. but they are really useful to have an idea about the nature of the problem.

There are many more methods of mathematical proof.

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You should try to see if Kevin Houston's How to Think Like a Mathematician can help.

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Here's one more strategy. ${}$

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Haha, It is the most followed strategy in the world. – user249332 Mar 6 at 10:02

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