Reddit mentions of Discrete Mathematics: Elementary and Beyond (Undergraduate Texts in Mathematics)

(Note: I wrote this elsewhere)

Discrete Mathematics. It teaches the basics of the following 5 key concepts in theoretical computer science:

• function
  
• set
  
• recursion
  
• proposition
  
• proof
  
  When you master these concepts, you will see that all hairy, formal definitions boil down to functions, sets, and propositions (with quantifiers). Recursion appears in many interesting structures in computer science, as well as in proofs of theorems (which are themselves propositions).
  
  I don't know of any good online introduction to discrete math, but here are a few book ideas.
  
  
• For a good textbook on discrete mathematics, check Schaum's Outline of Discrete Mathematics. It's also cheap.
  
• For a less textbook-ish overview written in an entertaining manner, Discrete Mathematics: Elementary and Beyond is my personal favourite.
  
• If you want to learn-by-doing, then The Haskell Road to Logic, Maths and Programming teaches you discrete math by having you manipulate discrete math concepts in the programming language Haskell. Here is a good free online Haskell tutorial.

My favorite introductory discrete math textbook is https://www.amzn.com/0387955852. (It also appears to be available for less unreasonable prices.)

this book really helped me in undergrad. Has a lot of really good concepts. It went along with a course but it does a great job on its own explaining some of the most relevant concepts to computer science.

Discrete mathematics and any proof based math in general is what college based math should be like- if you continue to take upper level math and CS courses, you will undoubtedly face this style of math again. Plug and chug (which is what a lot of calculus is) will no longer be the norm.

There is often a very large learning curve for students who are not used to seeing this type of math- so don't stress out too much about it. Eventually, you'll break a point where everything will make (sort of) sense. I went through the exact same thing when I took discrete for the first time, and I felt like I was getting destroyed on everything (I still suck at some topics) until I suddenly hit a point of clarity where I could see how most topics were tied in together. Mathematics, and especially an introductory discrete course, is cruel in that way- that every topic you learn is inherently related to each other, so if you already fall behind just a little, the mountain to catch up just becomes incredibly massive incredibly fast- and it's hard to even pinpoint a place to even start to catch up.

You may be lost in learning elementary proof techniques, or number theory, and then the next topic (say it's graph theory) utilizes a bunch concepts and previous proofs from number theory, and then the next topic might use something proved in graph theory and number theory, and so on. All of a sudden, nothing makes sense, and to learn topic ___, you need to know graph theory, but to know graph theory, you need to know number theory, but you don't know number theory that well, and some topics in number theory can perhaps be explained by another topic in graph theory (or any topic for that matter) The chain is all interlinked and it may difficult to even see where to start- but it is for this reason that once you cross this steep barrier, most things will suddenly become clear to you.

So I'd advise you to just continue visiting professor office hours, asking more questions, asking for other students' help, doing more and more practice. It may seem like you're getting nowhere, but you're essentially learning a new language right now, so it'll obviously take sometime until you feel as if you know what you're doing. Figuring out where people get the intuition to suggest seemingly random functions or a set of numbers or some assumption will come to you slowly, and slowly you'll break more and more of this chain.

https://www.amazon.com/Discrete-Mathematics-Laszlo-Lovasz/dp/0387955852 is another book my professor enjoyed using as a supplmenet.

I'd recommend Discrete Mathematics, Elementary and Beyond By Lovász, Pelikán, and Vesztergombi. It's the book I'm using in my undergraduate discrete math course, and I think it's a great introductory book that explores many areas of discrete math, and should allow you to see which field interests you most.

Here are some books I'd recommend.

General Books

These are general books that are more focused on proving things per se. They'll use examples from basic set theory, geometry, and so on.

1. How to Prove It: A Structured Approach by Daniel Velleman
   
2. How to Solve It: A New Aspect of Mathematical Method by George Pólya
   
   Topical Books
   
   For learning topically, I'd suggest starting with a topic you're already familiar with or can become easily familiar with, and try to develop more rigor around it. For example, discrete math is a nice playground to learn about proving things because the topic is both deep and approachable by a beginning math student. Similarly, if you've taken AP or IB-level calculus then you'll get a lot of out a more rigorous treatment of calculus.
   
   

• An Invitation to Discrete Mathematics by Jiří Matoušek and Jaroslav Nešetřil
  
• Discrete Mathematics: Elementary and Beyond by László Lovász and Jaroslav Pelikan
  
• Proofs from THE BOOK by Martin Aigner and Günter Ziegler
  
• Calculus by Michael Spivak
  
  I have a special place in my hear for Spivak's Calculus, which I think is probably the best introduction out there to math-as-she-is-spoke. I used it for my first-year undergraduate calculus course and realized within the first week that the "math" I learned in high school — which I found tedious and rote — was not really math at all. The folks over at /r/calculusstudygroup are slowly working their way through it if you want to work alongside similarly motivated people.
  
  General Advice
  
  One way to get accustomed to "proof" is to go back to, say, your Algebra II course in high school. Let's take something I'm sure you've memorized inside and out like the quadratic formula. Can you prove it?
  
  I don't even mean derive it, necessarily. It's easy to check that the quadratic formula gives you two roots for the polynomial, but how do you know there aren't other roots? You're told that a quadratic polynomial has at most two distinct roots, a cubic polynomial has a most three, a quartic as most four, and perhaps even told that in general an n^(th) degree polynomial has at most n distinct roots.
  
  But how do you know? How do you know there's not a third root lurking out there somewhere?
  
  To answer this you'll have to develop a deeper understanding of what polynomials really are, how you can manipulate them, how different properties of polynomials are affected by those manipulations, and so on.
  
  Anyways, you can revisit pretty much any topic you want from high school and ask yourself, "But how do I really know?" That way rigor (and proofs) lie. :)