Reddit reviews A Transition to Advanced Mathematics

I enjoyed the class. The professor was awesome, so that helped. I thought it was pretty easy, but I think that was because I had already been introduced to proofs. We did some Number Theory, Set Theory, Counting, Relations, Modular Arithmetic, Functions, Limits, Axiom of Choice and the Cantor-Schroder-Bernstein Theorem. We spent roughly two weeks on each subject, so we didn't go too in depth. At the end, we did some combinatorics because the professor likes combinatorics.
The book we used was A Transition to Advanced Mathematics by Douglas Smith. I didn't really use it at all. Our notes were sufficient.

I definitely think introduction to proof classes are helpful (and fun), but I would rather the school recommend a book to read over the summer so there is more room for another math elective. Naturally, this depends on the motivation of the school's students. My school has a bunch of lazy blobs. I doubt more than 5 would read a book over the summer.

This is the book I’m using right now in my first proofs class it’s pretty good at explaining the thought processes as well as it can be paired with How to Prove It by Daniel J. Velleman for a more through brake down of them problem types.

Just as an add-on, Stewart's is definitely the best way to go for learning applied calculus as a beginner. It's EXHAUSTIVE, though I'd actually recommend the full-on "Calculus" textbook instead of "Early Transcendentals" or "Single (Multi) Variable" texts for this reason:

At the end of every chapter, there are "problems plus" that will really challenge the way you think about what you've just learned. You don't get these in the other books. They'll make you think like a mathematician or a scientist instead of a "plug-and-chugger."

And once again, I'm gonna plug Smith's "Transition to Advanced Mathematics" for an introduction to proof-writing and set theory and the most basic of analysis. http://www.amazon.com/Transition-Advanced-Mathematics-Douglas-Smith/dp/0495562025/ref=sr_1_1?ie=UTF8&qid=1371247275&sr=8-1&keywords=douglas+smith+transition

Though definitely get an older edition to save more money. And I realize you can't get books delivered--you can find pdf versions of older editions.......

As for the lower, pre-calculus stuff, just look to the right on this reddit for Khan Academy and just browse through the topics there. If you're as good a student as you say you are, you just need the few holes filled in and a quick refresher, and Khan is perfect for it.

Check out A Transition to Advanced Mathetmatics. I took an enjoyable course with this book before starting to get deeper into my career and it was a nice primer.

Edit: .pdf version.

> 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.

>It gives me that mental stimulation I desire and that I feel I am genuinely am good at and don't need to have talent for because no matter what, so long as I put in the effort, then I got it down.

That's exactly the right attitude to have. :)

If I can make a recommendation, pick yourself up a copy of "A Transition to Abstract Mathematics" or a similar text and start working your way through it. You start with logic tables and learn about set theory. You'll enjoy it if you are interested in the "whys" of math, and if you end up picking math as a major, it will be helpful stuff to review ahead of time.

I recommend purchasing yourself a copy of this book: https://www.amazon.com/Transition-Advanced-Mathematics-Douglas-Smith/dp/0495562025

Chapter 0 is especially great, as it guides you through some of the basic grammar of mathematics. Most of the material is seen in some form or another in 115A(H), but I personally found this book to be a much better introduction to the upper division courses.

Isn't that true of any subject one likes?
Regardless, besides the Linear Algebra textbook, here are some books you should look at as well. These should give you a taste of what your introductory classes might be:

http://www.amazon.in/Transition-Advanced-Mathematics-Survey-Course/dp/0195310764

http://www.amazon.in/Transition-Advanced-Mathematics-Douglas-Smith/dp/0495562025

PM me if you want pdfs.

OK I see your point about the need to retain order for addition and subtraction. I wasn't thinking about that. The part that really gets me though is the second part, because the Wikipedia entry on ordered pairs doesn't say anything about addition/subtraction; that was part of this formula, not the definition of ordered pairs itself. It just says a nested tuple is equal to a flat tuple.

So I can see the need to retain order for this, but if the function were defined as this:

> MULT : R X R -> R

Then the output would be MULT(a, b) and order would not be important.

I just don't really see what is inherent in a Cartesian product operation that forces it to result in a flat tuple vs nested tuples.

But then again, like I said that is my programming background speaking, so maybe I'm missing something.

Thanks for taking a stab at it anyway. I may bring this up again sometime if I stumble over another problem like this.

EDIT I think I just realized why R X R creates a set of ordered pairs. I was flipping through the "look inside" of this book at Amazon and came across a fact I had failed to recall when looking at the function notation: R X R is the establishment of a 2-space Cartesian plane of the reals, therefore all objects resulting from R X R are points referenced by ordered x/y pairs. In the same way, R X R X R is the creation of a Cartesian 3-space of the reals, all of whose mathematical objects consist of points referenced by 3-tuples of x/y/z coordinates. And so on for R^n etc.

This stuff gets hard. :\

I still haven't taken calculus. I think that is my next step (well, after stats), then maybe that book, it looks oustanding.

All I know is that they're no longer doing Fitzpatrick or Chartrand (according to what a professor told me). Here's the new book. I think it's possible the course will be less analysis-focused. I think they should incorporate some abstract algebra into it. This goes into effect next semester by the way.