Reading, Writing, and Proving

by Daepp, Ulrich; Gorkin, Pamela

• ISBN13: 9780387008349
• ISBN10: 0387008349
• Format: Hardcover
• Copyright: 2003-08-01
• Publisher: Springer Verlag

Summary

This book, which is based on Pòlya's method of problem solving, aids students in their transition from calculus (or precalculus) to higher-level mathematics. The book begins by providing a great deal of guidance on how to approach definitions, examples, and theorems in mathematics. It ends by providing projects for independent study. Students will follow Pòlya's four step process: learn to understand the problem; devise a plan to solve the problem; carry out that plan; and look back and check what the results told them. Special emphasis is placed on reading carefully and writing well. The authors have included a wide variety of examples, exercises with solutions, problems, and over 40 illustrations, chosen to emphasize these goals. Historical connections are made throughout the text, and students are encouraged to use the rather extensive bibliography to begin making connections of their own. While standard texts in this area prepare students for future courses in algebra, this book also includes chapters on sequences, convergence, and metric spaces for those wanting to bridge the gap between the standard course in calculus and one in analysis.

Table of Contents

Preface	vii
1 The How, When, and Why of Mathematics	1	(16)
Spotlight: George Pólya	10	(4)
Tips on Doing Homework	14	(3)
2 Logically Speaking	17	(14)
3 Introducing the Contrapositive and Converse	31	(8)
4 Set Notation and Quantifiers	39	(14)
Tips on Quantification	51	(2)
5 Proof techniques	53	(10)
Tips on Definitions	61	(2)
6 Sets	63	(16)
Spotlight: Paradoxes	72	(7)
7 Operations on Sets	79	(10)
8 More on Operations on Sets	89	(8)
9 The Power Set and the Cartesian Product	97	(12)
Tips on Writing Mathematics	106	(3)
10 Relations	109	(10)
Tips on Reading Mathematics	115	(4)
11 Partitions	119	(10)
Tips on Putting It All Together	127	(2)
12 Order in the Reals	129	(18)
Tips: You Solved It. Now What?	144	(3)
13 Functions, Domain, and Range	147	(16)
Spotlight: The Definition of Function	157	(6)
14 Functions, One-to-One, and Onto	163	(12)
15 Inverses	175	(16)
16 Images and Inverse Images	191	(16)
Spotlight: Minimum or Infimum	199	(8)
17 Mathematical Induction	207	(16)
18 Sequences	223	(14)
19 Convergence of Sequences of Real Numbers	237	(14)
20 Equivalent Sets	251	(10)
21 Finite Sets and an Infinite Set	261	(10)
22 Countable and Uncountable Sets	271	(12)
23 Metric Spaces	283	(14)
24 Getting to Know Open and Closed Sets	297	(16)
25 Modular Arithmetic	313	(18)
26 Fermat's Little Theorem	331	(12)
Spotlight: Public and Secret Research	337	(6)
27 Projects	343	(36)
Tips on Talking about Mathematics	343	(3)
27.1 Picture Proofs	346	(2)
27.2 The Best Number of All	348	(2)
27.3 Set Constructions	350	(3)
27.4 Rational and Irrational Numbers	353	(2)
27.5 Irrationality of e and π	355	(3)
27.6 When Does f -1 = 1/f?	358	(2)
27.7 Pascal's Triangle	360	(2)
27.8 The Cantor Set	362	(4)
27.9 The Cauchy-Bunyakovsky-Schwarz Inequality	366	(2)
27.10 Algebraic Numbers	368	(3)
27.11 The RSA Code	371	(3)
Spotlight: Hilbert's Seventh Problem	374	(5)
28 Appendix	379	(4)
28.1 Algebraic Properties of R	379	(1)
28.2 Order Properties of R	380	(2)
28.3 Pólya's List	382	(1)
References	383	(6)
Index	389