The Art of Proof

Basic Training for Deeper Mathematics

  • Matthias Beck
  • Ross Geoghegan

Part of the Undergraduate Texts in Mathematics book series (UTM, volume 0)

Table of contents

  1. Front Matter
    Pages i-xxi
  2. The Discrete

    1. Front Matter
      Pages 1-1
    2. Matthias Beck, Ross Geoghegan
      Pages 3-12
    3. Matthias Beck, Ross Geoghegan
      Pages 13-23
    4. Matthias Beck, Ross Geoghegan
      Pages 25-31
    5. Matthias Beck, Ross Geoghegan
      Pages 33-45
    6. Matthias Beck, Ross Geoghegan
      Pages 47-54
    7. Matthias Beck, Ross Geoghegan
      Pages 55-64
    8. Matthias Beck, Ross Geoghegan
      Pages 65-72
  3. The Continuous

    1. Front Matter
      Pages 73-73
    2. Matthias Beck, Ross Geoghegan
      Pages 75-83
    3. Matthias Beck, Ross Geoghegan
      Pages 85-93
    4. Matthias Beck, Ross Geoghegan
      Pages 95-105
    5. Matthias Beck, Ross Geoghegan
      Pages 107-112
    6. Matthias Beck, Ross Geoghegan
      Pages 113-119
    7. Matthias Beck, Ross Geoghegan
      Pages 121-129
    8. Matthias Beck, Ross Geoghegan
      Pages 131-131
  4. Further Topics

    1. Front Matter
      Pages 133-133
    2. Matthias Beck, Ross Geoghegan
      Pages 135-139
    3. Matthias Beck, Ross Geoghegan
      Pages 141-144

About this book

Introduction

The Art of Proof is designed for a one-semester or two-quarter course. A typical student will have studied calculus (perhaps also linear algebra) with reasonable success. With an artful mixture of chatty style and interesting examples, the student's previous intuitive knowledge is placed on solid intellectual ground. The topics covered include: integers, induction, algorithms, real numbers, rational numbers, modular arithmetic, limits, and uncountable sets. Methods, such as axiom, theorem and proof, are taught while discussing the mathematics rather than in abstract isolation. Some of the proofs are presented in detail, while others (some with hints) may be assigned to the student or presented by the instructor. The authors recommend that the two parts of the book -- Discrete and Continuous -- be given equal attention. The book ends with short essays on further topics suitable for seminar-style presentation by small teams of students, either in class or in a mathematics club setting. These include: continuity, cryptography, groups, complex numbers, ordinal number, and generating functions.

Keywords

Cardinal number Countable set algebra cardinality completeness of R high-order recursions integers modulo n noncomputable numbers set theory strong induction universal quantifiers well ordering principle

Authors and affiliations

  • Matthias Beck
    • 1
  • Ross Geoghegan
    • 2
  1. 1., Department of MathematicsSan Francisco State UniversitySan FranciscoUSA
  2. 2.State University of New York, Department of Mathematical SciencesBinghamton UniversityBinghamtonUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4419-7023-7
  • Copyright Information Matthias Beck and Ross Geoghegan 2010
  • Publisher Name Springer, New York, NY
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-1-4419-7022-0
  • Online ISBN 978-1-4419-7023-7
  • Series Print ISSN 0172-6056
  • About this book