A Transition to Abstract Mathematics

A Transition to Abstract Mathematics

Learning Mathematical Thinking and Writing

2nd Edition - September 4, 2008

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  • Author: Randall Maddox
  • Hardcover ISBN: 9780123744807
  • eBook ISBN: 9780080922713

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Constructing concise and correct proofs is one of the most challenging aspects of learning to work with advanced mathematics. Meeting this challenge is a defining moment for those considering a career in mathematics or related fields. A Transition to Abstract Mathematics teaches readers to construct proofs and communicate with the precision necessary for working with abstraction. It is based on two premises: composing clear and accurate mathematical arguments is critical in abstract mathematics, and that this skill requires development and support. Abstraction is the destination, not the starting point.Maddox methodically builds toward a thorough understanding of the proof process, demonstrating and encouraging mathematical thinking along the way. Skillful use of analogy clarifies abstract ideas. Clearly presented methods of mathematical precision provide an understanding of the nature of mathematics and its defining structure. After mastering the art of the proof process, the reader may pursue two independent paths. The latter parts are purposefully designed to rest on the foundation of the first, and climb quickly into analysis or algebra. Maddox addresses fundamental principles in these two areas, so that readers can apply their mathematical thinking and writing skills to these new concepts. From this exposure, readers experience the beauty of the mathematical landscape and further develop their ability to work with abstract ideas.

Key Features

  • Covers the full range of techniques used in proofs, including contrapositive, induction, and proof by contradiction
  • Explains identification of techniques and how they are applied in the specific problem
  • Illustrates how to read written proofs with many step by step examples
  • Includes 20% more exercises than the first edition that are integrated into the material instead of end of chapter


upper level undergraduate mathematics students
Mathematicians and computer science professionals.

Table of Contents

  • Notation and Assumptions

    Section I: Foundations of Logic and Proof Writing Ch 1. Logic
    Ch 1. Language and Mathematics
    Ch 2. Properties of Real Numbers
    Ch 3. Sets and Their Properties
    Ch 4. Functions

    Section II: Basic Principles of Analysis
    Ch 5. The Real Numbers
    Ch 6. Sequences of Real Numbers
    Ch 7. Functions of a Real Variable

    Section III: Basic Principles of Algebra
    Ch 6. Groups
    Ch 7. Rings



Product details

  • No. of pages: 384
  • Language: English
  • Copyright: © Academic Press 2008
  • Published: September 4, 2008
  • Imprint: Academic Press
  • Hardcover ISBN: 9780123744807
  • eBook ISBN: 9780080922713

About the Author

Randall Maddox

Affiliations and Expertise

Pepperdine University, Malibu, CA, USA

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  • ShahenAlexanian Wed Jul 31 2019

    Excellent for University Preparation

    This is an excellent book for anyone interested in bridging to a university mathematics program.