Reddit reviews How to Prove It: A Structured Approach

I can't think of anything that's more important in math than proofs. Study a subject that involves a lot of proofs (any advanced math, really) such as linear algebra or analysis, and practice. How to Prove It by Velleman may help you get started. Writing proofs is just applying logic, definitions, and previously proved theorems.

I just started the PhD program this semester at North Carolina State. The program in general isn't ranked well but I'm interested in Environmental and Resource Econ and NCSU is top 10 (arguably top 5) in that field. I thought I'd give you a brief overview of the math that I had to prepare (undergrad rather than a math camp).

1. 3 semester's of calculus and diff eq - Really important for anything you're going to do in terms of optimization.
   
2. Linear Algebra - Important for econometrics stuff. Most applied stuff is easy enough in Stata but most programs will make you derive everything.
   
3. Real Analysis (lower) - I had an intro level class that went over set theory stuff as well as techniques needed to prove a statement. I would highly recommend an intro to proving course. If you're looking to study on your own I would suggest this book.
   
4. Real Analysis (upper) - My other RA courses involved deriving the real numbers, proving calculus, continuity proofs, etc. It's good in terms of practicing methods of proof but the material itself isn't great. That said, an A in RA is a great signal for grad schools. Anything lower than a B+ starts to get uncomfortable.
   
5. Topology - Some schools like to see it but no one is expecting it.
   
6. Optimization Theory - A course is unnecessary but its a good idea to look over primal/dual theory.
   
7. Probability Theory - You should, in my opinion, know the cute probability stuff front and back. Make sure to be familiar with compound events and whatnot. A probability class will probably get into random variables towards the end and those turn out to be very important.
   
8. Statistics Theory - More stuff on random variables, transformations, and statistical inference. Very important but unless you want to do econometric theory I think you can get away without knowing testing methods.
   
   One big thing that I didn't work on was programming skills. If you are intending to do applied work rather than theory, you'll want to be a solid programmer. Matlab and/or Maple are valuable and Stata, SAS, and ArcMap don't hurt.
   
   That said, I've met a lot of people in decent PhD programs who do not have much more than Calc, diff EQ, and linear algebra. I don't know if they passed comps or not but they got in. There are a number of good programs ranked 50+ that will teach you the math needed for applied work. However, if you want to go to a top 20 program you should definitely look into a math undergrad.
   
   Good luck to anyone thinking about applying.

http://www.amazon.com/How-Prove-It-Structured-Approach/dp/0521446635

That's what I started with and it was very helpful. The next semester when I took abstract vector spaces (proof-based linear algebra) I found writing the proofs to generally be straightforward because I'd already learned how to write a proof.

Daniel J Velleman's How to Prove It : A Structured Approach


This book is a pretty dang good intro to proofs, I highly reccommend it. This is the first edition, so you'll be able to find a used copy for super cheap.

This turned into kind of a treatise, but you are in the same position I was once, so here goes...

First of all, this is about the best introduction to proofs you can get. It's $17. You should buy this now and read it. Do the problems, too - they're fun and not particularly hard.

As for other advice, if I were you, I'd just graduate so you have a bachelor's and then go back for pure math. That way if you don't end up liking it, at least you'll have something.

You could also just switch majors now if you're sure you want to do it, but take it from me, you're not going to do it in 2 years. The important thing is, even if you could, you wouldn't want to. If you're getting into pure math to go to graduate school, you need to keep in mind that your intense 2 years of studying is exactly what the rest of us do for 4 years. The minimum requirements for a math degree are exactly that - the bare minimum. In fact, I myself switched during the 4th year of an art degree, planning to graduate after 2 years, and am now at the tail end of my 3rd year and no longer have any intention to graduate "early." I'm just doing what I would have done if I had started in math normally, because I realized I want to be my best for graduate schools.

Point is, don't cheat yourself out of this by trying to get some fuckin BA in math. If you decide to do it, do it for real.

(Note: This is assuming you're looking for grad school. If your plan is to stop at bachelor's and then work, consider stats or applied math or double majoring math with something else, cause you ain't doin' shit with only a bachelor's in pure math. That's just a fact.)

This being said, the decision to become a mathematician is the best one I ever made. I was in your position and I am so much happier - even now, when all my old friends have graduated and I'm in "major switch purgatory" - than I would have been if I would have kept trying to be something I'm not. So, I'm not trying to be discouraging. It really is worth a thought.

Here is how you make the decision... next semester, find out if your university has a proofs class. It will probably be for sophomore mathematics majors and use a book similar to the one I linked. Take this class alongside whatever humanities requirements you'd be taking anyway. If it has prerequisites other than 1st year calc (it shouldn't), talk to the math advisor and get them waved. The class probably won't be very hard, but it will give you an idea of what the process of "doing a math problem" evolves into when you get to higher level math. After this, find an introductory abstract algebra class (not a linear algebra class - one that includes group theory), and an introductory analysis class. This way you'll get a taste of two very different flavors of upper level math, and you'll be able to see how doing proofs actually works out. If you find yourself wanting more, then switch (or graduate and go back). If you don't, then don't be a math major. All in all taking three classes is a pretty inexpensive way to find out whether you want to do something, and since you can fit them into your fourth year, it won't fuck up the option of graduating with cinema studies if you decide math isn't your thing.

Get a logic book. For math majors at my University Sets and Logic is required before Linear Algebra which is the first proof intensive class.

http://www.amazon.com/How-Prove-Structured-Daniel-Velleman/dp/0521446635

This is the textbook. Very helpful.

Set theory/proof-writing is much more difficult than high school algebra. I'm teaching myself Calc III and Proof Writing right now in preparation for Abstract Algebra - I can say that compared to Calculus, the more advanced set theory is much more difficult. For me, anyway. For the Proof Writing I am using this book - How To Prove It: A Structured Approach. I'd say looking through that before moving on to anything more advanced than Calculus is a good idea. Which.. is why I'm doing it, myself.

I know a few people who highly recommend How to Prove It by Velleman. I've never read it so I can't say for sure. The first book I used to learn mathematical logic was Lay's Analysis with an Intro to Proof. I can't recommend that book enough. The first quarter of the book or so is a pretty gentle introduction to mathematical logic, sets, functions, and proof techniques. I imagine it will get you where you need to be pretty quickly.

I've been taking it this year, and we've been using Velleman's "How to Prove It." Unfortunately, there aren't answers for all the problems, but I've found it to be a pretty good book. Amazon

A lot of early math tends to come down to how often you do problems, and computation classes can generally be seen as rote learning. I'd suggest you start doing some proofs, they force you to understand what you are doing, rather then just doing what you've seen. Pick up http://www.amazon.com/How-Prove-It-Structured-Approach/dp/0521446635

or, http://www.amazon.com/Calculus-4th-Michael-Spivak/dp/0914098918/ref=sr_1_1?s=books&ie=UTF8&qid=1345011596&sr=1-1&keywords=spivaks+calculus

Not knowing random operations as you listed is fine, with time you will get quicker, but don't worry if you need to consider it for a moment.

How to Read and Do Proofs by Daniel Solow this book saved my life in Abstract Algebra.

I can't really give a better testimonial other than I read through this book and applied a couple of the concepts and did very well in the course.

One thing to remember, you can always reverse your steps, if you are stuck at some point, then work backwards from the end and you can sometimes meet up to the point you were stuck at.

Also, How to Prove It by Daniel J Velleman is another classic book that can help.