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© 2003

Proofs and Fundamentals

A First Course in Abstract Mathematics

Textbook

Table of contents

  1. Front Matter
    Pages i-xxi
  2. Proofs

    1. Front Matter
      Pages 1-1
    2. Ethan D. Bloch
      Pages 3-54
    3. Ethan D. Bloch
      Pages 55-103
  3. Fundamentals

    1. Front Matter
      Pages 105-105
    2. Ethan D. Bloch
      Pages 107-133
    3. Ethan D. Bloch
      Pages 135-176
    4. Ethan D. Bloch
      Pages 177-201
    5. Ethan D. Bloch
      Pages 203-248
  4. Extras

    1. Front Matter
      Pages 249-249
    2. Ethan D. Bloch
      Pages 251-321
    3. Ethan D. Bloch
      Pages 323-362
    4. Ethan D. Bloch
      Pages 363-373
  5. Back Matter
    Pages 375-424

About this book

Introduction

In an effort to make advanced mathematics accessible to a wide variety of students, and to give even the most mathematically inclined students a solid basis upon which to build their continuing study of mathematics, there has been a tendency in recent years to introduce students to the for­ mulation and writing of rigorous mathematical proofs, and to teach topics such as sets, functions, relations and countability, in a "transition" course, rather than in traditional courses such as linear algebra. A transition course functions as a bridge between computational courses such as Calculus, and more theoretical courses such as linear algebra and abstract algebra. This text contains core topics that I believe any transition course should cover, as well as some optional material intended to give the instructor some flexibility in designing a course. The presentation is straightforward and focuses on the essentials, without being too elementary, too exces­ sively pedagogical, and too full to distractions. Some of features of this text are the following: (1) Symbolic logic and the use of logical notation are kept to a minimum. We discuss only what is absolutely necessary - as is the case in most advanced mathematics courses that are not focused on logic per se.

Keywords

abstract mathematics adopted-textbook NY ksa logic proofs algebra cardinality Counting Division Equivalence function homomorphism linear algebra Mathematica mathematical induction mathematical proof mathematics Permutation proof

Authors and affiliations

  1. 1.Department of Mathematics and Computer ScienceBard CollegeAnnandale-on-HudsonUSA

Bibliographic information

  • Book Title Proofs and Fundamentals
  • Book Subtitle A First Course in Abstract Mathematics
  • Authors Ethan D. Bloch
  • DOI https://doi.org/10.1007/978-1-4612-2130-2
  • Copyright Information Birkhäuser Boston 2003
  • Publisher Name Birkhäuser Boston
  • eBook Packages Springer Book Archive
  • Hardcover ISBN 978-0-8176-4111-5
  • Softcover ISBN 978-1-4612-7426-1
  • eBook ISBN 978-1-4612-2130-2
  • Edition Number 1
  • Number of Pages XXI, 424
  • Number of Illustrations 0 b/w illustrations, 0 illustrations in colour
  • Topics Mathematical Logic and Foundations
    Mathematics, general
  • Buy this book on publisher's site
Industry Sectors
Finance, Business & Banking

Reviews

". . . Proofs and Fundamentals has many strengths. One notable, strength, is its excellent organization. The book begins with a three-part preface, which makes its aims very clear. There are large exercise sets throughout the book . . . Exercises are well integrated with the text and vary appropriately from easy to hard . . . Topics in Part III are quite varied, mostly independent from each other, and truly dependent on Parts I and II. At the end of the book there are useful hints to selected exercises.

Perhaps the book’s greatest strength is the author’s zeal and skill for helping students write mathematics better. Careful guidance is given throughout the book. Basic issues like not abusing equal signs are treated explicitly. Attention is given to even relatively small issues, like not placing a mathematical symbol directly after a punctuation mark. Throughout the book, theorems are often followed first by informative ‘scratch work’ and only then by proofs. Thus students can see many examples of what they should think, what they should write, and how these are usually not the same."

–MAA Online

"This is a well-written book, based on very sound pedagogical ideas. It would be an excellent choice as a textbook for a 'transition' course." ---Zentralblatt Math