Reddit reviews Discrete Mathematics, 2nd Edition

Here's my rough list of textbook recommendations. There are a ton of Dover paperbacks that I didn't put on here, since they're not as widely used, but they are really great and really cheap.

Amazon search for Dover Books on mathematics

There's also this great list of undergraduate books in math that has become sort of famous: https://www.ocf.berkeley.edu/~abhishek/chicmath.htm

Pre-Calculus / Problem-Solving

• The Art of Problem Solving - Lehoczky/Ruczyk - in two volumes, both have full solutions available separately
  
  Calculus
  
  
• Calculus - Stewart - ignore the bad reviews, it's very clear and covers all of basic calculus well - it's just the book that most intro students use and as a consequence gets a bad rap
  
  
• Calculus - Spivak - this will take your understanding of calculus to the next level
  
  
• Calculus - Apostol - there are two volumes and they are very comprehensive, albeit difficult
  
  Linear Algebra
  
  
• Linear Algebra - Lay
  
  
• Linear Algebra Done Right - Axler
  
  Differential Equations
  
  
• Elementary Differential Equations - Boyce/DePrima
  
  
• Ordinary Differential Equations - Tenenbaum
  
  Number Theory
  
  
• Number Theory - Pommersheim for a change of pace and proof skills
  
  Proof-Writing
  
  
• Transition to Higher Mathematics - Dumas/McCarthy - helps to bridge the gap between "lower" and higher math
  
  Analysis
  
  
• The Way of Analysis - Strichartz
  
• Principles of Mathematical Analysis - Rudin
  
• Real and Complex Analysis - Rudin
  
  Complex Analysis
  
  
• Complex Analysis - Ahlfors
  
  Functional Analysis
  
  
• Functional Analysis - Rudin
  
  
• Introductory Functional Analysis with Applications - Kreyszig
  
  Partial Differential Equations
  
  
• Partial Differential Equations for Scientists and Engineers - Farlow
  
  Higher-dimensional Calculus and Differential Geometry
  
  
• Calculus on Manifolds - Spivak - for a modern take on multivariable calculus
  
  
• Differential Geometry - Kreyszig
  
  Abstract Algebra
  
  
• A First Course in Abstract Algebra - Fraleigh
  
• Abtract Algebra - Dummit and Foote
  
  Geometry
  
  
• The Elements - Euclid - for culture and for understanding where math has come from, plus it's still an awesome book after all these years; don't need to go through all of it
  
• Euclidean and Non-Euclidean Geometries - Greenberg
  
  Topology
  
  
• Introduction to Topology - Mendelson
  
  
• Topology - Munkres
  
  Set Theory and Logic
  
  
• Introduction to Set Theory - Hrbacek/Jech
  
  
• Elements of Set Theory - Enderton
  
  
• Mathematical Logic - Kleene
  
  
• Set Theory and the Continuum Hypothesis - Cohen
  
  Combinatorics / Discrete Math
  
  
• Combinatorics - Cameron
  
  
• Concrete Math - Graham/Knuth/Patashnik
  
  
• Discrete Math - Biggs
  
  Graph Theory
  
  
• A First Course in Graph Theory - Chartrand/Zhang
  
  P. S., if you Google search any of the topics above, you are likely to find many resources. You can find a lot of lecture notes by searching, say, "real analysis lecture notes filetype:pdf site:.edu"
  

It sounds like you might perhaps want a background in Number Theory and/or Basic Logic and/or Set Theory. The thing about math is that there is a lot...

My advice for a text that might serve you well is N.L. Biggs' Discrete Mathematics (http://www.amazon.com/Discrete-Mathematics-Norman-L-Biggs/dp/0198507178). If you are at all interested in computer science, this is also a great book for that because it introduces some of the mathematical rigor behind it. Some people have a smidgen of difficulty with this text because it doesn't give some names to proofs/algorithms that maybe you've heard whispered (e.g. Dijkstra's shortest path and Prim's minimal spanning tree). A text that I tend to think is on par with Biggs', but many think is vastly superior (I love both, but for different reasons) that covers some (most) of the same topics is Eccles' An Introduction to Mathematical Reasoning (http://www.amazon.com/Introduction-Mathematical-Reasoning-Peter-Eccles/dp/0521597188/ref=pd_sim_b_4?ie=UTF8&refRID=1BB6VKRP59S2420M132F). This book has a wonderful focus on building from the ground up and emphasizes clearly worded and mathematically rigorous proofs.

You seem genuinely interested in mathematics, but I do want to warn you about some more ahem esoteric (read: improperly worded, perhaps?) problems that ask such things as why 1 is greater than 0. The mathematics here is largely armchair - lacking any fundamental logic. There would be no issue with redefining a set of bases such that "0" is greater than "1". However, if you want to have rationale of the concept of things being greater than another, that's more like number theory. You can learn the 10 axioms of natural numbers and then build from there.

Both of the books I mentioned will cover stuff like this. For example, they both (unless I'm not remembering correctly) delve into Euclid's proof of infinite primes, something which may interest you.

Briefly (and not so rigorously), assume that the number of primes, p1, p2, p3, ..., pN, is finite. Then there exists a number P which is the product of these primes. Based on the axioms of natural numbers, since all primes p1,p2,...,pN are natural numbers P is a natural number and so is P+1. Consider S = P+1. If S is prime than our list is incomplete, assume S isn't prime. Then some number in our list, say pI, divides S because any natural number can be written as the product of primes. pI must also divide P because P equals the sum of all primes. Therefore if pI divides S and pI divides P, then pI divides S-P = 1. That's a contradiction because no prime evenly divides 1.

Stuff like this is super cool, super simple, and super beautiful and you absolutely can learn it. These two books would be a great place to start.

There are books designed for the IB qualification

https://www.amazon.co.uk/Mathematics-Higher-Diploma-Option-Discrete/dp/1107666945/ref=sr_1_2?ie=UTF8&qid=1521592587&sr=8-2&keywords=a-level+discrete+maths

Or I read this and it was good

https://www.amazon.co.uk/Discrete-Mathematics-Norman-L-Biggs/dp/0198507178/ref=sr_1_3?ie=UTF8&qid=1521592587&sr=8-3&keywords=a-level+discrete+maths

that's kinda ironic.. was about to start my search

Discrete Mathematics
by Norman L. Biggs 2nd.


https://www.amazon.ca/Discrete-Mathematics-Norman-L-Biggs/dp/0198507178