该书对初学高等数学的读者来说特别有意义。它的基本出发点是引导读者以研究的心态去学习，让读者养成独立思考的习惯，并进而成为研究型的学习者。该书将一个学习数学的过程在某种意义下程序化，努力让学习者养成一个好的学习习惯，以及学会如何应对问题。该书特色鲜明，类似的图书确实很少。

本书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。

本书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。

本书对初学高等数学的读者来说特别有意义。

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 Fermat's Little Theorem

27 Projects

 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 ∏

 27.6 When Does f-1=1/f?

 27.7 Pascal's Triangle

 27.8 The Cantor Set

 27.9 The Cauchy-Bunyakovsky-Schwarz Inequality

 27.10 Algebralc Numbers

 27.11 The RSA Code

 Spotlight: Hilbert's Seventh Problem

28 Appendix

 28.1 Algebraic Properties of R

 28.2 Order Properties of R

 28.3 P61ya's List

References

Index