Can Someone Teach Me How to Prove Theorems?

[Jamey]

I have a simple question that probably has a complicated answer. How do I prove theorems? Like, prove that the square root of 2 is irrational (or other things). I mean, I can do some geometric ones, but the symbolic ones confuse me. It makes sense when I read one, but I can't do one myself! I'll never be a good mathematician/physicist if I can't prove things! Help? Guidance? Anyone?

 

[Jorriss]

There's no set method. There's a variety of techniques one may use - for proving sqrt(2) is irrational, for example, one generally goes for proof by contradiction. You may want to look at a book on mathematical reasoning or proofs. Velleman, Eccles, etc. There are many options.

 

[slider142]

Simple mathematical propositions are usually of the form "Object X has property Y". Sometimes, you will have a list of one or more precepts, which creates a conditional statement, such as "If X is true, then Y is true."
In the former case, the proof relies on your being able to display the fact that object X does indeed possesses property Y. In this case, you want to be able to show that the square root of 2 has the property called "is an irrational number". Your first job, then, is to find out what it means for a number to be an irrational number.
In your research, you may find that one definition is that it cannot be written in the form of a fraction p/q where p and q are integers. So you would start this particular proof by trying to show that the square root of 2 cannot be written like this.
To do this, you now have to find a concrete definition of the object called "the square root of 2". One useful definition of this object is that it is the positive real number whose square is the number 2. So it is the positive number that solves the equation x² = 2.
Now we know we want x to not be a number of the form p/q, as mentioned above. So let us see if forcing x to be in the form p/q makes anything break. Thus, we start by assuming that there are indeed two integers p/q such that (p/q)² = 2. We are hoping that algebraic manipulation of this hypothetically false equation will lead us to an equation that is more obviously false. That is, we want to show it is equivalent to a more obviously false statement, such as a statement that directly contradicts an axiom of our number system, or contradicts a well established theorem.
Using algebra, and a little arithmetic, we quickly come to the conclusion that this equation can never be true for any pair of integers p and q. Thus, we see that if x is the square root of 2, it can never be represented in this way, and thus x cannot be a rational number. This property fits the definition of x being an irrational number, so our proof is completed.
As you can see, at the simpler stages of mathematical proofs, our job consists mostly of matching definitions, and showing that either a definition fits an object, or it does not fit.
However, it is not always easy to see how to apply a set of definitions to an object, or see how a more complicated conditional set of properties might fit together. There are many ways to tackle any given problem, subject to how a person sees the world. Most of the time spent on a proof should be in trying to understand the problem, which is finding out how the pieces fit together. You cannot start a proof if you do not understand the objects that are being manipulated and how they are related. It is only after seeing the connection that any proof should be attempted. It may sometimes fail, but you will usually be able to see then exactly why it failed.
For more in-depth advice on solving problems of a mathematical nature, see Polya's How To Solve It.

 

[Fredrik]

There are several books on the subject, for example "How to prove it" by Daniel Velleman, and "Book of proof" by Richard Hammack. I think they all include the basics of set theory and logic. It would be a good start to get one of these books, read a little, and do some exercises. You should post a few of your attempts in the homework forum, so that we can point out mistakes and suggest improvements.

Once you get a little experience, there are a few things that will feel so obvious to you that you will have a hard time understanding how it wasn't obvious to you before. For example:

1. If you're supposed to prove that every "fnurgle" has a certain property, then you have to use the definition of "fnurgle" somewhere in your proof. If you do a calculation that seems to prove the theorem without using the definition, then you need to think about whether you have used a theorem that was proved using the definition, or if you have just made a huge blunder. Usually it's the latter. If your argument is solid, but doesn't use the definition of "fnurgle", then you have proved a theorem about something other than fnurgles ("Fnurgle" is a nonsense word I just made up. You can substitute any mathematical term for it, like "continuous function" or "odd integer").

2. Every variable must be assigned a value, or be the target of a "for all" or "there exists". A proof consists of statements, not formulas whose truth values depend on the value of some variable. If you haven't assigned a value to x, a formula like $x^2=1$ isn't a statement (it's neither true nor false) and has no business appearing in your proof, other than as part of a "for all" or a "there exists" statement, like "there exists a real number x such that $x^2=1$".

3. A lot of theorems are "for all" statements. For example, "for all x in A, x has property P". The proof should almost always start with "let x be an arbitrary element of A", and then you try to prove that this x has property P. The reason is simply that this is how you split a "for all" statement into several sentences. This is almost always necessary, since very few proofs can be written as a single sentence that's short enough to be understood by the reader.

 

[disregardthat]

Look up ZFC on wikipedia. With those axioms and modus ponens you should be good to go.
Joking aside... there's not really a set of rules to follow. My advice would be to go through simple proofs step by step, and then trying to prove some simple things yourself. Fredrik's suggestions are good, you might want to get yourself a book like that. If you really want to get good at proofs, I suggest looking into olympiad-like problems. Try "How to solve it" by Polya, or "Art and Craft of problem solving" by Zeitz. There you will get as difficult problems as you could possibly want as exercises, which is a great way to learn different proof techniques.