How to Think Like a Mathematician by Houston

[micromass]

• Author: Kevin Houston
• Title: How to Think Like a Mathematician: A Companion to Undergraduate Mathematics
• Amazon Link: https://www.amazon.com/dp/052171978X/?tag=pfamazon01-20
• Prerequisities: High-School Mathematics
• Level: Undergrad

Table of Contents:

[LIST]
[*] Preface
[*] Study skills for mathematicians
[LIST]
[*] Sets and functions
[*] Reading mathematics
[*] Writing mathematics I
[*] Writing mathematics II
[*] How to solve problems
[/LIST]
[*] How to think logically
[LIST]
[*] Making a statement
[*] Implications
[*] Finer points concerning implications
[*] Converse and equivalence
[*] Quantifiers - For all and There exists
[*] Complexity and negation of quantifiers
[*] Examples and counterexamples
[*] Summary of logic
[/LIST]
[*] Definition, theorems and proofs
[LIST]
[*] Definitions, theorems and proofs
[*] How to read a definition
[*] How to read a theorem
[*] Proof
[*] How to read a proof
[*] A study of Pythagoras' Theorem
[/LIST]
[*] Techniques of proof
[LIST]
[*] Techniques of proof I: Direct method
[*] Some common mistakes
[*] Techniques of proof II: Proof by cases
[*] Techniques of proof III: Contradiction
[*] Techniques of proof IV: Induction
[*] More sophisticated induction techniques
[*] Techniques of proof V: Contrapositive method
[/LIST]
[*] Mathematics that all good mathematicians need
[LIST]
[*] Divisors
[*] The Euclidean Algorithm
[*] Modular arithmetic
[*] Injective, surjective, bijective - and a bit about infinity
[*] Equivalence relations
[/LIST]
[*] Closing remarks
[LIST]
[*] Putting it all together
[*] Generalization and specialization
[*] True understanding
[*] The biggest secret
[/LIST]
[*] Appendices
[LIST]
[*] Greek alphabet
[*] Commonly used symbols and notation
[*] How to prove that ...
[/LIST]
[*] Index
[/LIST]

 

[mathwonk]

i have not read this, but if you want a free taste, here it is:

http://www1.maths.leeds.ac.uk/~khouston/pdf/10ways.pdf