Reddit reviews Mathematical Thinking: Problem-Solving and Proofs (2nd Edition)

Unfortunately for you, 251 learning is mostly from lecture and recitation lessons, for which there is not an official textbook (student informally use the Concepts of Mathematics Textbook, which is quite decent).

This is the public course website: https://colormygraph.ugrad.cs.cmu.edu/15251-s12/

Course materials are located in the calendar tab and many of them are public.

You will have a tough time learning anything of consequence without something like videos of the lectures, etc. (Which even students don't have access to)

For proofs in general, I like D'Angelo and West's Mathematical Thinking. http://www.amazon.com/Mathematical-Thinking-Problem-Solving-Proofs-Edition/dp/0130144126

For discrete math, especially combinatorics, I loved Miklos Bona's A Walk Through Combinatorics. http://www.amazon.com/Walk-Through-Combinatorics-Introduction-Enumeration/dp/9814335231/

For induction proofs, you check your base case, assume the induction hypothesis (true for k), and then check k+1.

You should be able to manipulate the k+1 term into something involving the k term, and that will then lead to the k+1 conclusion.

Example For all n >= 4, 2^(n) < n!

Base case: n = 4. 2^(4) = 16 < 24 = 4!

IH: Assume true for some k >= 4.

Then 2^(k+1) = 2*2^(k)

2*2^(k) < 2*k! (Induction Hypothesis used here)

2*k! < (k+1)k! (k > 3, so k+1 > 2)

(k+1)k! = (k+1)! (definition of factorial)

I like "Mathematical Thinking." You can get the PDF online quite easily

http://www.amazon.com/gp/aw/d/0130144126

I used Mathematical Thinking: Problem-Solving and Proofs by D’Angelo and West, and I remember it being quite a good book as an introduction to proofs. We didn’t use the book extensively in that course, but when we did need it I had no complaints.

I'd start with a discrete math course (often offered for intro computer-science, but make sure the curriculum doesn't consist of any coding). Then move on to real analysis.


I really like this book as an intro.

> My problem is that I have never really been introduced to sets or other things,

How did that happen? I know that at my alma mater, you're supposed to have some kind of proofs-oriented course before you take intro abstract algebra (either "abstract linear algebra", which is a proofs-heavy intro to linear algebra, or "fundamental mathematics" or "theory of computation"). Does this course not have appropriate prereqs, or did you disregard them?

Edit: the text that the fundamental mathematics class there uses is Mathematical Thinking: Problem-Solving and Proofs. It's written by a couple of the professors from the university. I don't know much about West, but I had D'Angelo for real analysis, and he was both meticulous and clear in lecture; I'd be surprised if any book that he put his name on was not.

> The set/subset relation could be considered an inverse relation as well.

Let A = {1, 2} be a set. Then B = {1} is a subset of A. Let's define a relation between them, f: A -> B given by f(1) = 1 and f(2) = 1. f is, actually, a function. But this function f doesn't have an inverse. Why? Find out from Mathematical Thinking: Problem-Solving and Proofs by D'Angelo and West.

Therefore

> Then that would also make the infinite/finite relation an inverse relation.

doesn't follow.

Ah yes, traditionally math learning is a fairly linear progression and is bottlenecked up until you take your first proof/analysis class, after which your path can branch out. Seeing as how you already have a link there, a textbook is listed for that class and that one happens to be popular so maybe you can buy that one. Me personally, I used this one when I went through the fundamentals

We are using Mathematical Thinking: Problem Solving & Proofs 2nd Edition. We get lectures notes because the text book is difficult to understand, but they dont really help..

EDIT: I realize now its the same book! Great! Any help?