How can I improve my proof-writing skills?

[john562]

Don't get how to write proofs!

I'm a high school student who really wants to major in mathematics. I love reading proofs, but when the book [What is Mathematics by Courant] asks me to do proofs, I have absolutely no idea of where to start. Should I just give up my aspiration to major in math because I for the love of god can't figure out how to write a proof.

 

[Char. Limit]

A better thing to do would be to ask someone for help on how to write a proof. This site is a very good resource for that. What problems in particular are you having?

 

[john562]

Alright. I'm actually working on it right now, but will start a new thread when I'm done.

 

[gb7nash]

A classic beginners proof is to show that an odd number times an odd number is an odd number. That might be a good one to try.

 

[micromass]

The problem with beginners is that they are often scared of the word "proof". They think it is far more rigourous than it actually is. In fact, a proof is simply an argument why something is true. Instead of asking yourself the question "Prove that...", you might ask yourself the question "Why is...". This is thesame thing.

If I ask you, "why does the game of tic-tac-toe always end in a draw if the pllayers are good enough", then you will immediately start reasoning with "let's say player 1 puts an x there, and player 2 puts an o there, then..." Now you're actually doing a proof, but you don't realize it.

When trying to prove things, it is important to read a lot of proofs and to understand them well. A lot of proofs will contain the same idea and once you grasp this idea, you're settled. Proofs, like anything, require experience.

And you shouldn't drop you're aspirations of mathematics just because you have troubles with proofs now. I struggled over 2 years (of self-study in high-school) with writing proofs, and now I find it fun!

As a last suggestion: buy a good proof-book and read it cover-to-cover...

 

[I like Serena]

Learning to write a (proper) proof is like learning a new language.
The difference is that at least the words and grammar are not so ambiguous and full of exceptions.
Anyway, it takes a while to learn it and get comfortable with it.

 

[dodo]

If it helps (it does for me), a good starting point towards a proof is to make inventory of (a) what you have, what is given by the problem statement, (b) where you want to arrive. A proof is just a bridge between these two; you'd be surprised of how often people have troubles that spawn simply from a light understanding of (or a quick jump over) the statement of the question.

 

[dalcde]

It is difficult to have one single formula to prove everything. There are several methods (actually many) to prove things. The simplest way is to do some algebra. For example, if we want to prove that (a+b)(a-b)=a²-b², you expand the left hand side and get to the right hand side. Then you proved that the equation (technically speaking, it's an identity) above is correct. There are many methods and you have to learn them one py one.

 

[Rasalhague]

One thing that can make me seize up on seeing a proof question is not knowing what the game rules are: what are the axioms, what can be assumed. But I think this instinct of knowing what the question is asking for is something that comes with experience.

For example, I remember puzzling over some questions in Binmore's Mathematical Analysis, only to turn to the answers in the back and find that he'd taken for granted certain preliminaries that I'd thought would have to be part of the proof. By contrast, Spivak's Calculus covers some of the same material, at a similar level, but makes explicit what axioms can be used. I think the latter approach motivates me to try harder to solve the problem, knowing that it can be solved given what I know, and that it will be satisfying to have the thing proved right back to the foundations. Whereas Spivak gives all the axioms, Binmore gives only some rules and says something along the lines of the usual rules of arithmetic are assumed - even though some rules one might consider usual are among the things to be solved. For this reason, a book like Spivak's, which makes the task clear, may be the best starting point in attempting proofs. That said, Binmore's answers are often very detailed, so they give plenty of good, worked examples of proofs, which is useful too.

Of course, a more advanced book on a topic will rely on elementary results; then the reader has to use their own judgement about what can be reasonably assumed. Books will sometimes set an easier or more casual exercise with the words, "The reader should convince themselves." In other words, assume whatever seems obvious to you; just consider what's new and not quite obvious until it becomes so; if it seems to make sense then, you've probably got it.

In his Real Analysis lectures, Francis Su advises students to write their proofs in a way that assumes as much knowledge as someone two or three weeks behind them in the course.

It's interesting the contrast between micromass's answer and that of I like Serena. micromass emphasises the instinctive aspect (we're writing proofs all the time in our heads without knowing it), whereas I like Serena stresses the need to study mathematical logic as if it was a foreign language. Not a contradiction, just an interesting difference in emphasis... Does anyone have any recommendations for a book on proofs or mathematical logic? Am I right in thinking that the official name of this language is first order logic, or is it predicate logic in Wikipedia's more general sense? Or is it predicate logic in the sense of their second paragraph: the idea which can be formalized as first order logic (although first order logic is not necessarily the only way to formalize it)?

 

[jobsism]

What's the procedure for proving something like the four-colour theorem? I read that for the proof, all possible types of figures were assimilated, and then eliminated...but when u see the question for the first time, where does one begin?

 

[micromass]

What's the procedure for proving something like the four-colour theorem? I read that for the proof, all possible types of figures were assimilated, and then eliminated...but when u see the question for the first time, where does one begin?

Buy a book on graph theory or discrete mathematics. For example in the book "Discrete mathematics" by Grimaldi, they prove the 5-colour theorem. Which is a weaker statement than the four-colour theorem as it states that any map can be coloured in by 5 colours!

The proof is quite easy and it can show you how to prove things like this!

 

[jobsism]

Thanks, micromass! It's tough getting books like these where I live, but I'll try anyway!:)

 

[micromass]

Check out this website: http://hbpms.blogspot.com/2008/05/stage-1-introductory-discrete.html

It contains many books in discrete mathematics. I guess some of these will contain a proof of the five-color theorem. Look for sections on "planar graphs".