Reading, Writing, and Proving: A Closer Look at Mathematics

2nd Edition (Undergraduate Texts in Mathematics)

Ulrich Daepp, Pamela Gorkin
2.ª edição

Springer | 2011 | 391 páginas | pdf | 5 Mb

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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 and ends with suggested projects for independent study. Students will follow Polya's four step approach: analyzing the problem, devising a plan to solve the problem, carrying out that plan, and then determining the implication of the result. In addition to the Polya approach to proofs, this book places special emphasis on reading proofs carefully and writing them well. The authors have included a wide variety of problems, examples, illustrations and exercises, some with hints and solutions, designed specifically to improve the student's ability to read and write proofs. 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.

Contents
Chapter 1: The How, When, and Why of Mathematics
Solutions to Exercises
Spotlight: George Pólya
Chapter 2: Logically Speaking
Chapter 3: Introducing the Contrapositive and Converse
Chapter 4: Set Notation and Quantifiers
Chapter 5: Proof Techniques
Chapter 6: Sets
Spotlight: Paradoxes
Chapter 7: Operations on Sets
Chapter 8: More on Operations on Sets
Chapter 9: The Power Set and the Cartesian Product
Tips on Writing Mathematics
Chapter 10: Relations
Tips on Reading Mathematics
Chapter 11: Partitions
Tips on Putting It All Together
Chapter 12: Order in the Reals
Chapter 13: Consequences of the Completeness of R
Tips: You Solved It. Now What?
Chapter 14: Functions, Domain, and Range
Spotlight: The Definition of Function
Chapter 15: Functions, One-to-One, and Onto
Chapter 16: Inverses
Chapter 17: Images and Inverse Images
Spotlight: Minimum or Infimum?
Chapter 18: Mathematical Induction
Chapter 19: Sequences
Chapter 20: Convergence of Sequences of Real Numbers
Chapter 21: Equivalent Sets
Chapter 22: Finite Sets and an Infinite Set
Chapter 23:Countable and Uncountable Sets
Chapter 24: The Cantor–Schröder–Bernstein Theorem
Spotlight: The Continuum Hypothesis
Chapter 25:Metric Spaces
Chapter 26: Getting to Know Open and Closed Sets
Chapter 27: Modular Arithmetic
Chapter 28: Fermat’s Little Theorem
Spotlight: Public and Secret Research
Chapter 29: Projects
Tips on Talking about Mathematics
29.1 Picture Proofs
Guided Project
Open-Ended Project
Notes and Sources
29.2 The Best Number of All (and Some Other Pretty Good Ones)
29.3 Set Constructions
29.4 Rational and Irrational Numbers
29.5 Irrationality of e and p
29.6 A Complex Project
29.7 When Does f-1 = 1/f ?
29.8 Pascal’s Triangle
29.9 The Cantor Set
29.10 The Cauchy–Bunyakovsky–Schwarz Inequality
29.11 Algebraic Numbers
29.12 The Axiom of Choice
29.13 The RSA Code
Spotlight: Hilbert’s Seventh Problem
Appendix
Algebraic Properties of R
Order Properties of R
Axioms of Set Theory
Pólya’s List
References
Index