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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
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).
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
Content Level »Lower undergraduate
Keywords »Cantor's theorems - Fundamental Theorem of Arithmetic - counting principles - mathematical induction - number-theoretic functions - proof techniques - relations and partitions - set constructions - transition course