Where can I find challenging proof problems to practice?

[AlexVGheo]

I really like maths, and I really like doing mathematical problems.I have learned pretty much everything I need to know for the next year of the school syllabus already by just reading up on stuff that interested me i.e. calculus, complex analysis etc. but that's all about solving stuff and finding things. What I really want to do is proofs but I don't know anythings which I don't either already know and understand a proof off or is too simple it isn't challenging to prove i.e. 1*0=0 or is too hard and out of my league for the moment i.e. Fermat's little theorem [although that is pretty high up! it is just very difficult to think of an example]

So the question is really if anyone can recommend an entry level place to find proof problems?

 

[Theorem.]

Learning how to work through proofs and the different techniques of proof takes time,
thankfully there are many resources available. To start there is a great book introducing techniques of proof with many questions also: http://www.amazon.com/dp/0521675995/?tag=pfamazon01-20.
In fact, many elementary introductory algebra books, geometry, etc have extensive practice problems involving proofs, and such books aren't too hard to get your hands on (Basically any introductory rigorous mathematics textbook). You can also look at Mathematical Olympiad problems and Putnam problem. I'll post some specific resources if any come to mind.

EDIT:
Here is a database of mathematical olympiad problems: http://www.imo-official.org/problems.aspx

Any mathematics students should start with the proofs of the irrationality of the square root of two and the proof that there are infinitely many primes (Try these!).

 

[dustbin]

I don't think the suggestion for olympiad and putnam problems is very good. These are generally incredibly challenging and not a good place to begin learning how to prove things.

Although there are some good books on learning proofs, I don't think they are necessary. In fact, I suggest not reading them and just jumping into a proof-based math text (which one depends on your background). Take a shot at the problems and find someone to critique your work (perhaps here in the homework help section). Revise your proof based on the criticism(s) until you get it right. In my opinion, this is the best way to learn how to prove things. You'll get a general idea of proof methods from a book, but you won't really learn how to prove things until you try to do so yourself and then mess up. Moreover, you have to learn when a certain proof method should be applied... something that can't really be taught in a book.

I enjoyed Allendoerfer & Oakley's Principles of Mathematics. You will learn logic and set theory, about groups and fields, and review/expand upon many concepts you have seen previously (analytic geometry, calculus, etc.). I think this is a very good bridge to proof-based math that is much better than a "learn proofs" book.

I think the book "An Introduction to Inequalities" by Beckenbach and Bellman would be great for learning how to do proofs. You will have some familiarity with the material, but it will solidify (and extend) your understanding of inequalities. This will come in handy when you take real analysis. As well, the authors spell out the proofs and somewhat outline what it takes to prove something. I really like this book.

My other recommendation would be Apostol's or Spivak's Calculus. I think Apostol would be somewhat gentler if you are just learning proofs. The reading is more terse, but I feel the problems start out a bit easier and then gradually increase with difficulty.

 

[Theorem.]

Although the putnam is definitely something he'd have to work his way up to, I strongly disagree in regards to the olympiad. The olympiad has a wide range of problems, and not all are difficult. Additionally there is a huge database of problems, many of which are accessible. It is also intended for a high school audience. That doesn't mean that will be the place to start but it is definitely a place to grow in skill and confidence. Another good starting point would be any elementary book on graph theory. Most of the proofs in elementary graph theory are induction or basic proof by contradictions, and it is often taught to a wider audience.

 

[Theorem.]

Although there are some good books on learning proofs, I don't think they are necessary. In fact, I suggest not reading them

Some of the books actually provide a strong introduction to logic. Which can be VERY necessary.
Although taking questions from a variety of books can be a good idea it will also be very time consuming and perhaps inaccessible if the goal is to learn simply proving techniques and getting comfortable with them (although indeed it will help!). The book I suggested is a good precursor that could be even read alongside a book that simply offers problems. I really think the logic background is important and that is what the book mostly focuses on. It also has A LOT of questions in it for practice.

 

[AlexVGheo]

Thank you both =)
I think I will have a look at the Olympiad problems and see if I am indeed capable of doing them and then follow your advice Dustbin and check the ones I am uncertain about here.

As for the books I appreciate that learning by book methods isn't a great way to become good and creative at proving and I assure you I don't wish to memorise 'forms' of argument. All I really want is as Theorem said a database of problems I can try my hand at, but it is meaningless unless I can check my answer and I don't wish to pester the forum with every one in case I have a myriad of them

Great advice, thank you

 

[dustbin]

Thank you both =)
I don't wish to pester the forum with every one in case I have a myriad of them

That's why we're here :-)

 

[Theorem.]

Yes I would take dustbin's advice though he is definitely right that a lot of the problems may be too intimidating at first! just work your way towards them. There are hundreds of problems there that will really help you develop problem solving and proof skills
!

 

[1MileCrash]

If you are just starting proofs (which I gathered from your post) it is also important to practice coherent and concise proof writing with the proper etiquette. You do this by practicing on simple proofs and learning techniques (direct, contradiction, contrapositive, induction) and it is also important to learn about logic, negations, etc.

It is possible to know exactly how to prove some theorem perfectly, and then write an awful proof.

 

[verty]

I think set theory is a great way to practice proofs. As you come to each theorem or lemma, hide the proof and try to prove it yourself, then compare your completed proof to the given one, think about other proof ideas, etc.

 

[Theorem.]

@1milecrash the etiquette you speak of comes directly from the underlying logic- which is indeed important to understand. But the logic and techniques are actually equivalent- they aren't too different things. I agree that logic is important though- which is why i suggested a book that includes several chapters on logic. Learning symbolic logic in general can be useful. It really helped me understand the structure of proofs. As far as the details go- that comes with practice : )

 

[1MileCrash]

Etiquette such as defining your terms, stating your assumptions, justifying any claims you make, stating your goals, and using good "mathematical English" (as my professor called it) obviously have little to do with pure logic, and much more to do with "penmanship."

 

[Theorem.]

The things you mentioned during Your original post had everything to do with logic. Believe it or not the language of mathematics is logic so what you just mentioned has lots to do with logic:)