Becoming Fluent in Math Proofs - Tips & Advice

[IKonquer]

I'm starting to learn how to write proofs, and I am wondering how to become fluent in proofs. Is it necessary to do problems that are IMO/Putnam? Can anyone give me some advice?

Thanks in advance.

 

[micromass]

I'm starting to learn how to write proofs, and I am wondering how to become fluent in proofs. Is it necessary to do problems that are IMO/Putnam? Can anyone give me some advice?

Thanks in advance.

Finding and writing proofs is a very slow process. And no, you don't need to be able to do Putnam problems, instead I suggest you start easy.

Take a subject that you would like to learn (or one that you already had a class on). I (or somebody else) can probably recommend good books on the subject, so feel free to ask. Study the proofs very carefully, then close the book and try to replicate the proof. Make sure you understand EVERYTHING, every little detail counts.
Then, try to make proofs that do not differ much from the proofs you just replicated. Try to make exercises that use the same little dirty tricks. You'll learn quickly what the tricks are.

And (here comes to most important part): present every proof you make to somebody knowledgeable so (s)he can comment on it and present ways on doing things cleaner/nicer/better. You don't need to bother your prof with it, just posting it here on PF is already fine.

Don't try to do the hard things right away, you need experience with the method and with the tricks to be able to write proofs. Don't be discouraged if you have to spend weeks proving silly things like "the sum of two even numbers is even" (this is an exageration, but hey).

 

[IKonquer]

Hi micromass, thanks for the advice. I'm going to be taking abstract algebra next semester and I have had some exposure to the ideas of proof by induction, contradiction, truth tables, and the like in high school. But I've never taken an introduction to proofs class in college. Do you think it is possible to learn straight from Fraleigh's abstract algebra w/o having a formal course on proofs? I think my major problem is when I don't follow proofs or understand each step when books give examples.

 

[micromass]

Hi micromass, thanks for the advice. I'm going to be taking abstract algebra next semester and I have had some exposure to the ideas of proof by induction, contradiction, truth tables, and the like in high school. But I've never taken an introduction to proofs class in college. Do you think it is possible to learn straight from Fraleigh's abstract algebra w/o having a formal course on proofs? I think my major problem is when I don't follow proofs or understand each step when books give examples.

In my opinion, abstract algebra is the ideal (no pun intended) course to start learning proofs! You may find it worthwhile to keep a proof book close to you (like Velleman's book). You don't need to read the entire proof book, but it might be handy to look something up.

Before you can start proving things on your own, it is essential to understand all the theory. So you must understand each step. If you don't understand a step, ask on physicsforums or ask your professor. Only when you understand the proof and are able to replicate the proof, can you start making exercises and start proving things for yourself.

Fraleigh's abstract algebra is a nice and easy book. I don't think you'll have troubles with it. Another great book is Pinter's "a book on abstract algebra".

I would recommend that you start learning from Fraleighs book this summer already. The sooner you're acquaneted with proofs, the better. It's better to hit roadblock on self-study, then during the year (where time is less of a luxury). And feel free to ask what you don't understand, and feel free to present proofs that you've made

 

[IKonquer]

What a coincidence! I actually have both Velleman's book and Abstract Algebra by Pinter. I actually started reading Velleman's book from the beginning and stopped. I didn't really understand the importance of unions, intersections, truth tables, and the whole operation on sets. Could you explain why they are important in mathematics?

Also how should one go about reading an abstract algebra book differently than a calculus book?

 

[micromass]

What a coincidence! I actually have both Velleman's book and Abstract Algebra by Pinter. I actually started reading Velleman's book from the beginning and stopped. I didn't really understand the importance of unions, intersections, truth tables, and the whole operation on sets. Could you explain why they are important in mathematics?

I like Velleman's book because it spends so much attention on unions, intersections and other set theoretic notions. I agree that Velleman's book isn't a book that you can read cover to cover, you must handle it like a reference work. If you don't understand something, then read the appropriate section in Velleman. Learning proofs while going through an abstract algebra course is better than learning proofs through a proof book!

Anyway, unions and intersections and stuff are important in mathematics because they form the language in which mathematics is written. Every mathematical theory after calculus will involve sets and will be written in the notation of set theory. So knowing how to work with sets is quite important.

For example, when writing down all the integers that are divisible by 3 and 4, we will write down

$$\{x\in \mathbb{Z}~\vert~3~\text{divides}~x\}\cap \{x\in \mathbb{Z}~\vert~4~\text{divides}~x\}$$

So sets are used here. Are sets essential here?? No, we could also write it down in an entirely different language, but it is the set theoretic language that we happen to use here.

Also how should one go about reading an abstract algebra book differently than a calculus book?

Depends on how you read your calculus book. I shouldn't make an example of me, but here's how I studied math texts:

- the first reading: read the importants texts, read the statements of the theorems, skip over the calculations. This should be quick reading.
- Work through the entire text. Make sure to understand everything you read (and make sure you understand why they do what they do). If you don't understand it, think about it. If the thinking takes too long, ask here.
- Take a piece of paper and try to write down the proofs and calculations that you've just studied. Only look in the book when you're really stuck. Repeat until you know and understand all the proofs.
- Try to expand on the theory: try to make examples of theorems, try to find counterexamples of when the theorem fails (for example, we require a certain number to be positive in the statement of the theorem? Try to find an example of when the theorem fails when the number is negative), make a mind-map of the chapter.
- Make exercises.

This is very time intensive, but it pays off!

 

[Oriako]

- the first reading: read the importants texts, read the statements of the theorems, skip over the calculations. This should be quick reading.
- Work through the entire text. Make sure to understand everything you read (and make sure you understand why they do what they do). If you don't understand it, think about it. If the thinking takes too long, ask here.
- Take a piece of paper and try to write down the proofs and calculations that you've just studied. Only look in the book when you're really stuck. Repeat until you know and understand all the proofs.
- Try to expand on the theory: try to make examples of theorems, try to find counterexamples of when the theorem fails (for example, we require a certain number to be positive in the statement of the theorem? Try to find an example of when the theorem fails when the number is negative), make a mind-map of the chapter.
- Make exercises.

This is quite possibly the most useful thing I've ever heard regarding studying from math textbooks.