© 2012

Introduction to Mathematical Structures and Proofs

  • Solutions manual for even numbered exercises is available on for instructors adopting the text for a course

  • Discusses the multifaceted process of mathematical proof by thoughtful oscillation between what is known and what is to be demonstrated

  • Presents more than one proof for many results, for instance for the fact that there are infinitely many prime numbers

  • Shows how the processes of counting and comparing the sizes of finite sets are based in function theory, and how the ideas can be extended to infinite sets via Cantor's theorems

  • Contains a wide assortment of exercises, ranging from routine checks of a student's grasp of definitions through problems requiring more sophisticated mastery of fundamental ideas

  • Demonstrates the dual importance of intuition and rigor in the development of mathematical ideas


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

Table of contents

  1. Front Matter
    Pages i-xiii
  2. Larry J. Gerstein
    Pages 1-35
  3. Larry J. Gerstein
    Pages 37-107
  4. Larry J. Gerstein
    Pages 109-140
  5. Larry J. Gerstein
    Pages 141-189
  6. Larry J. Gerstein
    Pages 191-275
  7. Larry J. Gerstein
    Pages 277-348
  8. Larry J. Gerstein
    Pages 349-373
  9. Back Matter
    Pages 375-401

About this book


As a student moves from basic calculus courses into upper-division courses in linear and abstract algebra, real and complex analysis, number theory, topology, and so on, a "bridge" course can help ensure a smooth transition. Introduction to Mathematical Structures and Proofs is a textbook intended for such a course, or for self-study.  This book introduces an array of fundamental mathematical structures. It also explores the delicate balance of intuition and rigor—and the flexible thinking—required to prove a nontrivial result.  In short, this book seeks to enhance the mathematical maturity of the reader.

 The new material in this second edition includes a section on graph theory, several new sections on number theory (including primitive roots, with an application to card-shuffling), and a brief introduction to the complex numbers (including a section on the arithmetic of the Gaussian integers). Solutions for even numbered exercises are available on for instructors adopting the text for a course. 

From a review of the first edition:

"...Gerstein wants—very gently—to teach his students to think. He wants to show them how to wrestle with a problem (one that is more sophisticated than "plug and chug"), how to build a solution, and ultimately he wants to teach the students to take a statement and develop a way to prove it...Gerstein writes with a certain flair that I think students will find appealing. ...I am confident that a student who works through Gerstein's book will really come away with (i) some mathematical technique, and (ii) some mathematical knowledge….

Gerstein’s book states quite plainly that the text is designed for use in a transitions course.  Nothing benefits a textbook author more than having his goals clearly in mind, and Gerstein’s book achieves its goals.  I would be happy to use it in a transitions course.”

—Steven Krantz, American Mathematical Monthly


Cantor's theorems Fundamental Theorem of Arithmetic counting principles mathematical induction number-theoretic functions proof techniques relations and partitions set constructions transition course

Authors and affiliations

  1. 1.University of California, Santa BarbaraSanta BarbaraUSA

About the authors

Larry Gerstein's primary areas of research have been in quadratic forms and number theory and he has published extensively in these areas. The author's first edition of "Introduction to Mathematical Structures and Proofs" has sold to date (8/2/2010) over 6000 copies and has gone through 5 printings. Gerstein himself has a transition course at UC, Santa Barbara (Math 8-A transition to higher mathematics) from his book since its first publication date. The first edition also received 2 glowing reviews by Steve Krantz for the American Mathematical Monthly, and S. Gottwald for Zentralblatt.

Bibliographic information

  • Book Title Introduction to Mathematical Structures and Proofs
  • Authors Larry J. Gerstein
  • Series Title Undergraduate Texts in Mathematics
  • Series Abbreviated Title Undergraduate Texts Mathematics
  • DOI
  • Copyright Information Springer Science+Business Media, LLC 2012
  • Publisher Name Springer, New York, NY
  • eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
  • Hardcover ISBN 978-1-4614-4264-6
  • Softcover ISBN 978-1-4939-5146-8
  • eBook ISBN 978-1-4614-4265-3
  • Series ISSN 0172-6056
  • Series E-ISSN 2197-5604
  • Edition Number 2
  • Number of Pages XIII, 401
  • Number of Illustrations 0 b/w illustrations, 0 illustrations in colour
  • Topics Mathematical Logic and Foundations
    Number Theory
  • Buy this book on publisher's site
Industry Sectors
Finance, Business & Banking