Contents
Preface Xi
1 The How，wnen. and Why of Mathematics
Spotlight： George P61ya
Tips on Doing Homework
2 Logically Speaking
3 Introducing the contrapositivc and Converse
4 Set Notation and opantificrs
Tips on Quantification
5 Proof Techniques
Tips on Definitions
6 Sets
Spotlight：Paradoxs
7 Operations on Sets
8 More on Operations on Sets
9 The Power Set and the Cartesian Product
Tips on Writing Mathematics
10 Relations
Tips on Reading Mathematics
11 Partitions
Tips on Putting It All Together
12 Order in the Reals
Tips：You Solved It Now What?
13 Functions， Domain， and Range
Spotlight：The Definition of Function
14 Functions， One-to-One， and Onto
15 Inverses
16 Images and Inverse Images
Spotlight. Minimum or Infimum
17 Mathematical Induction
18 Sequences
19 Convergence of Sequences of Real Numbers
20 Equivalent Sets
21 Finite Sets and an Infinite Set
22 Countable and Uncountable Sets
23 Metric Spaces
24 Getting to Know Open and Closed Sets
25 Modular Arithmetic
26 Fermats Little Theorem
Spotlight： Public and Secret Research
27 Projects
Tips on Talking about Mathematics
27.1 Picture Proofs
27.2 The Best Number of All
27.3 Set Constructions
27.4 Rational and Irrational Numbers
27.5 Irrationality of e and πr
27.6 When Doesf-1 = 1/f?
27.7 Pascals Triangle
27.8 The Cantor Set
27.9 The Cauchy-Bunyakovsky-Schwarz Inequality
27.10 Algebraic Numbers
27.11 The RSA Code
Spotlight： Hilberts Seventh Problem
28 Appendix
28.1 Algebraic Properties of R
28.2 Order Properties of R
28.3 Po1yas List
References
Index