Recommendation on books

Question

I am planning to finish a sequence of math books. I have finished requisite pre-calculus, trig-1 and have some experience in proofs. I am currently studying Hardy's
Course of Pure Mathematics(to learn calculus). Having my caliber defined, now I would like to have some books that include the following topics:
1.Solving mathematical problems using algorithms.Here is one on AM-GM inequality if you want to see an example:
2.Introducing Combinatorics, theory of functions and sets very rigorously.
3.Calculus books that contain a lots of good problems and practical applications so that I could keep touch with calculus.

Answer 1

Personal Favorites:

1. A Concise Introduction to Pure Mathematics by Martin Liebeck
2. Calculus by Gilbert Strang
3. The Art and Craft of Problem Solving by Paul Zeitz
4. Problem-Solving Through Problems by Loren Larson

Answer 2

Note that any answer will ultimately reflect the author's personal preference, so I will give mine, but please keep this in mind:

At the end of the day, mathematics should be about doing mathematics; a lot of the time this means doing problems, the rest of the time it will be about learning theory, that is learning its language, learning why the theory is phrased in the terms it is phrased and so on...

My suggestions will be leaning towards the pure mathematics side of things without many applications; I still hope you (or anyone reading this) find(s) this at least somewhat helpful:

• I think it is sensible to learn real analysis early on, a good start with lots of problems would be Rudin's Principles of mathematical analysis;
• A very beautiful theory to learn is algebra; there will be many definitions, lots of abstraction early on, but as soon as one gets accustomed to the type of proofs one faces in algebra, it is very satisfying to see how lots of problems have short, elegant solutions; hard to recommend any one book but Artin's Algebra is a classic (and it covers lots of linear algebra, which might be sensible) and if you want to go really abstract Aluffi's Algebra: Chapter 0 is absolutely beautiful;
• Since you asked about combinatorics and applications of 'calculus' I think it would be fun to learn some probability theory, here I think Jacod & Protter's Probability Essentials is great.

I am not qualified to give a suggestion for algorithmic mathematics, but I am sure some else is.

Considering some of the contents of my suggestions, one finds that some of them may seem very advanced for someone just starting to learn math. These recommendations are 'for the long run' rather than something one can finish in a short period of time. It seems, for example, entirely reasonable to me to work through some of the basics in Rudin, then jump to the very nice introduction to sets and categories in Aluffi, then to some linear algebra in Artin and so on. This then ultimately depends on your own preferences; you will find out what they are only by doing, so good luck!