数学的读写和证明

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

《数学的读写和证明》详细讨论数学的读写和证明，将学习数学的过程在某种意义下程序化，以使学习者培养出极好的学习习惯，以及学会如何应付面对的问题。《数学的读写和证明》努力将读者培养成独立思考研究型的学习者，并试图将这个过程程序化，同时通过大量的实例让读者接受这个程序。这也许代表了西方教育的一个特征，这个特征也许在中国是缺少的，在教科书中也是缺少的。 　　《数学的读写和证明》对初学高等数学的读者来说特别有意义。