Reading, Writing, and Proving: A Closer Look at Mathematics / Edition 2

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Overview

This book, which is based on Polya'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 Polya'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.

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Editorial Reviews

From the Publisher
From the reviews of the second edition:
“This is at heart a fairly conventional transition text, but it has a number of features that encourage students to practice and improve their mathematical reading, writing, and proving skills. … The book is primarily concerned with an exposition of those parts of mathematics in which students need a more thorough grounding before they can work successfully in upper-division undergraduate courses. … a mathematically-conventional but pedagogically-innovative take on transition courses.” (Allen Stenger, The Mathematical Association of America, September, 2011)
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Product Details

  • ISBN-13: 9781441994783
  • Publisher: Springer New York
  • Publication date: 6/29/2011
  • Series: Undergraduate Texts in Mathematics Series
  • Edition description: 2nd ed. 2011
  • Edition number: 2
  • Pages: 378
  • Sales rank: 843,692
  • Product dimensions: 6.20 (w) x 9.30 (h) x 1.10 (d)

Meet the Author

Ueli Daepp is an associate professor of mathematics at Bucknell University in Lewisburg, PA. He was born and educated in Bern, Switzerland and completed his PhD at Michigan State University. His primary field of research is algebraic geometry and commutative algebra.

Pamela Gorkin is a professor of mathematics at Bucknell University in Lewisburg, PA. She also received her PhD from Michigan State where she worked under the director of Sheldon Axler. Prof. Gorkin’s research focuses on functional analysis and operator theory.

Ulrich Daepp and Pamela Gorkin co-authored of the first edition of “Reading, Writing, and Proving” whose first edition published in 2003. To date the first edition (978-0-387-00834-9 ) has sold over 3000 copies.

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Table of Contents

Preface
1 The How, When, and Why of Mathematics 1
2 Logically Speaking 17
3 Introducing the Contrapositive and Converse 31
4 Set Notation and Quantifiers 39
5 Proof Techniques 53
6 Sets 63
7 Operations on Sets 79
8 More on Operations on Sets 89
9 The Power Set and the Cartesian Product 97
10 Relations 109
11 Partitions 119
12 Order in the Reals 129
13 Functions, Domain, and Range 147
14 Functions, One-to-One, and Onto 163
15 Inverses 175
16 Images and Inverse Images 191
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
27 Projects 343
28 Appendix 379
References 383
Index 389
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