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Reading, Writing, and Proving / Edition 1

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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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Product Details

  • ISBN-13: 9780387008349
  • Publisher: Springer-Verlag New York, LLC
  • Publication date: 8/28/2003
  • Series: Undergraduate Texts in Mathematics Series
  • Edition description: Older Edition
  • Edition number: 1
  • Pages: 408
  • Product dimensions: 6.20 (w) x 9.30 (h) x 0.90 (d)

Table of Contents

Preface
1 The How, When, and Why of Mathematics Spotlight: George Polya Tips on Doing Homework
2 Logically Speaking
3 Introducing the Contrapositive and Converse
4 Set Notation and Quantifiers Tips on Quantification
5 Proof Techniques Tips on Definitions
6 Sets Spotlight: Paradoxes
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 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 $\pi $
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 Algebraic Numbers
27.11 The RSA Code Spotlight: Hilbert's Seventh Problem
28 Appendix
28.1 Algebraic Properties of $\@mathbb {R}$
28.2 Order Properties of $\@mathbb {R}$
28.3 Polya's List References Index

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