数学的读写和证明 英文PDF电子书下载

The How,When,and Why of Mathematics 1

Spotlight:George Pólya 10

Tips on Doing Homework 14

2 Logically Speaking 17

3 Introducing the Contrapositive and Converse 31

4 Set Notation and Quantifiers 39

Tips on Quantification 51

5 Proof Techniques 53

Tips on Definitions 61

6 Sets 63

Spotlight:Paradoxes 72

7 Operations on Sets 79

8 More on Operations on Sets 89

9 The Power Set and the Cartesian Product 97

Tips on Writing Mathematics 106

10 Relations 109

Tips on Reading Mathematics 115

11 Partitions 119

Tips on Putting It All Together 127

12 Order in the Reals 129

Tips:You Solved It.Now What? 144

13 Functions,Domain,and Range 147

Spotlight:The Definition of Function 157

14 Functions,One-to-One,and Onto 163

15 Inverses 175

16 Images and Inverse Images 191

Spotlight:Minimum or Infimum 199

17 Mathematical Induction 207

18 Sequences 223

19 Convergence of Sequences of Real Numbers 237

20 Equivalent Sets 251

21 Finite Sets and an Infinite Set 261

22 Countable and Uncountable Sets 271

23 Metric Spaces 283

24 Getting to Know Open and Closed Sets 297

25 Modular Arithmetic 313

26 Fermat's Little Theorem 331

Spotlight:Public and Secret Research 337

27 Projects 343

Tips on Talking about Mathematics 343

27.1 Picture Proofs 346

27.2 The Best Number of All 348

27.3 Set Constructions 350

27.4 Rational and Irrational Numbers 353

27.5 Irrationality of e and π 355

27.6 When Does f-1=1/f? 358

27.7 Pascal's Triangle 360

27.8 The Cantor Set 362

27.9 The Cauchy-Bunyakovsky-Schwarz Inequality 366

27.10 Algebraic Numbers 368

27.11 The RSA Code 371

Spotlight:Hilbert's Seventh Problem 374

28 Appendix 379

28.1 Algebraic Properties of R 379

28.2 Order Properties of R 380

28.3 Pólya's List 382

References 383

Index 389