Reddit reviews Mathematical Proofs: A Transition to Advanced Mathematics (2nd Edition)

The rate of your learning is defined by your determination. If you don't give up then you will learn the material.

Look at the book that is required and only learn what you need in the class. Don't learn everything in the book either. Just learn what you need to do well and refer to the books when you get confused.

Note don't try to learn everything that's below. Only use it to learn what you actually need. This can be overwhelming at first but just set aside a set time to study this.
EDIT I added more books and courses.
OCW
http://ocw.mit.edu/courses/mathematics/18-01sc-single-variable-calculus-fall-2010/
http://ocw.mit.edu/courses/mathematics/18-02-multivariable-calculus-fall-2007/index.htm
http://ocw.mit.edu/courses/mathematics/18-03-differential-equations-spring-2010/
http://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/
Helpful books
http://www.amazon.com/Mathematical-Proofs-Transition-Advanced-Mathematics/dp/0321390539/ref=sr_1_3?s=books&ie=UTF8&qid=1312542911&sr=1-3
http://www.amazon.com/Understanding-Probability-Chance-Rules-Everyday/dp/0521540364
http://www.amazon.com/gp/product/048663518X/ref=pd_lpo_k2_dp_sr_1?pf_rd_p=486539851&pf_rd_s=lpo-top-stripe-1&pf_rd_t=201&pf_rd_i=0155510053&pf_rd_m=ATVPDKIKX0DER&pf_rd_r=0YXJR9EVHCH9PCBDN372

Khan Academy
http://khan-academy.appspot.com/#calculus
http://www.youtube.com/user/keithpeterb#p/u/19/dS2p_APpcnI
http://khan-academy.appspot.com/video/probability--part-1?playlist=Old%20Algebra
http://www.youtube.com/user/keithpeterb#p/u/19/dS2p_APpcnI
http://khan-academy.appspot.com/video/linear-algebra--introduction-to-vectors?playlist=Linear%20Algebra

EDIT: I knew nothing about topological quantum computation about 1.5 years ago but then I took a independent study in college and I was assigned 1-3 papers a week to read. Eventually I got it a few months ago. What got me through it was not giving up...

OCW

Single Variable Calculus

Multivariable Calculus

Differential Equasions

Linear Algebra

Helpful books

Mathematical Proofs: A Transition to Advanced Mathematics

Understanding Probability: Chance Rules in Everyday Life

Linear Algebra

Other Resources

Kahn Academy

Probability (keithpeterb on youtube)

Probability (Kahn)

Linear Algebra (Kahn)


There, Fixed up your links.

What is the "Highest Level" of mathematics you have taken?
Math is substantially more like a foreign language than popular culture would lead you to believe. It takes practice and what I like to call 'settle time.'
If you feel like you have a strong grasp on the concepts of algebra I highly recommend starting from 'scratch' (first principals) and getting a book like http://www.amazon.com/gp/offer-listing/0321390539/ref=sr_1_2_twi_har_1_olp?ie=UTF8&qid=1450533541&sr=8-2&keywords=mathematical+proofs

It was the first textbook that made me really start to understand what is needed to think like a mathematician. Start at the beginning work though problems, set theory is so much more important than most people realize. It will be cloudy and frustrating but really try to work some problems, put it down for a week let it stew and come back to the problems you had trouble with. Do that over and over.
While you are doing that pick up any elementary Stats/Prob and/or Linear Algebra book and start flipping through from the beginning you will see all the tools you are learning in Mathematical Proofs in those books as well. Try to take what you are learning and see it applied in those books to add some extra hooks to attach things to in your brain.

For Numerical Analysis you are going to want to build a strong base in proofs, linear algebra, set theory, and calculus as you go forward. Don't let this stop you from starting to read up it is a great way to stay excited when you are learning things to know fun ways that they are applied but don't get discouraged. My Numerical Analysis class was a Sr level college course that started the semester with 24 Math and CS majors about half gave up before the mid term/

During my sophomore year I took an "intro to proofs" course (known formally at the institution as Foundations of Advanced Mathematics) and I found it to be extremely beneficial in my development as a mathematician. We used Chartrand's "Mathematical Proofs" textbook (here's the link for those who are interested).

The text covered set theory, logic, the various proof methods, and then dug into stuff like elementary number theory, equivalence relations, functions, cardinality (culminating in Cantor's two main results), abstract algebra, and analysis. Obviously the book only scratched the surface on a lot of these topics, but I felt it accomplished its goal.

Part of my satisfaction with the course is likely due to the fact that we had a brilliant professor who taught the course in the spirit of what u/Rtalbert235 spoke of. He was able to clearly articulate the distinction between computation and theory. The way I like to say it is that he taught us the difference between pounding a bunch of nails into a 2X4 (computation) and building a house (proving theorems).

I don't mean to universally praise "intro to proofs" courses, however. I can definitely see how they can be horrible wastes of time if not done properly, and I can also appreciate the idea of "throwing" students into proof-based courses (analysis, algebra, and so on). For me though, I think it's worth the effort to try and optimize these sorts of classes, which will ultimately serve a LOT of math students who need to understand proofs, but don't necessarily have a desire to pursue the subject beyond the undergraduate level.

tl;dr - Given the right combination of textbook and professor, an "intro to proofs" course can be just what the doctor ordered for developing mathematicians.

I think "Mathematical Proofs: A Transition to Advanced Mathematics (2nd Edition)" is a solid book.

It starts off with what I would expect in a discrete math course (which is generally a first proofs course) and ends with a few chapters that would begin a second step writing intensive proofs course: number theory, calculus (real analysis), and group theory (algebra).

There are also many resources online that will help you once you've gotten through the basic notions in the book.

Mathematical Proofs: A Transition to Advanced Mathematics was the book for my college proofs class. I found it to be a good resource and easy to follow. It covers some introductory set theory as well. Just be prepared to work through the proof exercises if you really want a good intuition on the topic.

> Is there any good book with problems/examples that I could work through in order to thoroughly prepare myself to be able to write proofs for a Real Analysis I course?

Besides Velleman's "How to Prove it," try Mathematical Proofs: A Transition to Advanced Mathematics or maybe How to Read and Do Proofs: An Introduction to Mathematical Thought Processes.

The book I used in my "Intro to Proofs" course was A Transition to Advanced Mathematics. It was pretty good, but the edition that I used had several mistakes in it. Also, it's waaaay too expensive.

Now for the unpleasantries —

Suggestions aside, the main problem here is your "thoroughly prepare myself to be able to write proofs for Real Analysis" goal. Working through a proofs book on your own will be seriously challenging, but the thought of taking Real Analysis without at least two other proofs courses under your belt is terrifying to me. I had to take "An intro to mathematical proofs" followed almost immediately by a proof-based Linear Algebra course before I was even allowed to contemplate a Real Analysis course.

Come to think of it, how in the hell are you even allowed to do this if you haven't taken a proofs course before? Are you sure this is even possible? Are prerequisites not enforced at your school? No one, and I mean no one was permitted to take Abstract Algebra or Real Analysis without the required prerequisites at my university. The only way you could get around it was by being the next Andrew Wiles.

Just to drive all this home, I was a straight-A Physics/Math major with the exception of two courses: Thermodynamics and my first proofs course. I've never worked so damn hard for a B in my life. Come to think of it, I actually recall quantum mechanics being easier than my proofs course.

I'm being sincere when I ask you to reconsider this plan. You are asking for a world of pain followed by the very real possibility of failure if you do this.

TL;DR: Unless you are remarkably sharp and have loads of time on your hands, this is probably a mistake. You should take a more elementary proofs course before tackling Real Analysis. Good luck, whatever you choose to do.

One thing I like to remind people, is that Linear Algebra is really cool and though it tends to come "after" calculus for some reason, it really has no explicit calc prerequisite.

I highly recommend Dr. Gilbert Strang's lectures on it, available on youtube and ocw.mit.edu (which has problems, solutions, etc, also)

I think it's a great topic for right around late HS, early college. And he stresses intuition and imo has the right balance of application and theory.

I'd also say that contrary to most peoples' perceptions, a student's understanding of a math topic will vary greatly depending on the teacher. And some teachers will be better for some students, others for others. That's just my opinion, but I firmly believe it. So if you find yourself struggling with a topic, find another teacher/resource and perhaps it will be more clear. Of course this shouldn't diminish the effort needed on your part, learning math isn't a passive activity, one really has to do problems and work with the material.

And finally, proofs are of course the backbone of mathematics. Here is an intro text I like on that.

Oh okay, one more thing, physics is a great companion to math. I highly recommend "Classical Mechanics" by Taylor, in that regard. It will be challenging right now, but it will provide some great accompaniment to what you'll learn in upcoming years.

Don't think of your abilities as fixed. The number of proofs you encounter grows from where you are now. You did not know algebra when you started. You will be increasingly exposed to proofs as you go along. Spend time on them. I recommend you get a tutor, or at least read some extra material. https://www.amazon.com/Mathematical-Proofs-Transition-Advanced-Mathematics/dp/0321390539/ref=sr_1_69?ie=UTF8&qid=1494805054&sr=8-69&keywords=proofs+math

This may not be appropriate for you, but for someone else reading this thread, this book is great for learning about proofs, logic, and basic analysis techniques:

http://www.amazon.com/Mathematical-Proofs-Transition-Advanced-Mathematics/dp/0321390539/ref=sr_1_1?ie=UTF8&qid=1300643516&sr=8-1

Too bad they upped the price since I bought it. Nevertheless, one can probably find it used either on Amazon or Half.com

Having a proof explained to you isn't even close to the thrill of proving something yourself, IMO. My advice would be to get your hands on an intro to proofs text and work through some of it. If you don't like writing proofs or think it's boring, your time at university is probably going to bore you to tears.

If you want an intro proofs book, you might start here. The text is very clearly written and chapters 9-13 will give you a very basic notion of what ideas will be at the core of some of your upper-level math classes (abstract algebra, real analysis, etc).

In a math course I recently took that was basically an introduction to math proofs we used Mathematical Proofs: A Transition to Advanced Mathematics which I found to be a great text. It begins by going through the language and syntax used in proofs and slowly progresses through theory, different types of proofs, and eventually proofs from advanced calculus. There are so many examples that are very well laid out and explained. I would highly recommend it for learning proofs from scratch.

Possible path:

Learn to think like mathematicians because you'll need it. For example, Mathematical Proofs: A Transition to Advanced Mathematics by Chartrand et al is a good book for that. When you got the basics of math argumentation down, it's time for abstract algebra with emphasis on vector spaces(you really need good working knowledge of linear algebra). People like Axler's Linear Algebra Done Right. Maybe, study that. Or maybe work through Maclane's Algebra or Chapter 0 by Aluffi.

After that you want to get familiar with more or less rigorous calculus. One possibility is to study Spivak's Calculus, then pick up Munkres Analysis on Manifolds.

Up next: differential geometry which is your main goal. At this point your mathematical sophistication will have matured to the level of a grad student of math.

Good luck.