- #1

gstroot

- 26

- 1

I'm trying to self teach myself Fundamental Mathematics. I looked around, but I wasn't sure what to look for exactly. I read the part on Set Theory in "Book of Proof" by Richard Hammack. I enjoyed it, but I wasn't sure if it is rigorous enough to stand against a college level course.

The course I'm trying to mimic goes by this:

**Chapter 1: Numbers, Sets, and Functions (2 - 4 hours)**- Set-theoretic concepts and operations
- Functions: Formal definition and basic concepts (graph, image set, bounded function, decreasing/increasing function)
- Triangle and AGM inequalities

**Note:**Much of the material in Chapter 1 can be covered at a later point. For example, one could spend one or two hours on basic set-theoretic concepts, then move on to logical statements in Chapter 2, while deferring the formal definition of functions and related concepts to the beginning of Chapter 4.**Chapter 2: Langauge and Proof (4 - 6 hours)**- Logical statements, conditionals, quantifiers
- Methods of proof (direct, contraposition, contradiction)

**Chapter 3: Induction (4 - 6 hours)**- Sum/product notations
- Induction and strong induction
- Well-ordering principle
- Applications

**Chapter 4: Bijections and Cardinality (4 - 6 hours)**- More about functions: injective, surjective, bijective properties, compositions and inverses
- Cardinality, finite, countable, and uncountable sets
- Countability of rationals

**Chapters 13/14: Real Numbers, Sequences and Series (8 - 10 hours)**- Infinite sequences and series: Formal definition of convergence, basic properties of convergent sequences and series
- Sup, inf, and the Completeness Axiom
- Cauchy sequences and the Cauchy Convergence Criterion
- Montone Convergence Theorem
- Bolzano-Weierstrass Theorem

**Note:**These chapters introduce students to basic concepts in analysis and form an essential component of this course.

Obviously I could just their text, D'Angelo/West, Mathematical Thinking, Second Edition, but generally books recommended for classes aren't the best for self taught methods.

tl;dr best book for self teaching fundamental mathematics (Course of Topics above)