Introduction to Mathematical Structures and Proofs

This updated and revised second edition is designed to help students advance from basic calculus to higher-level linear and abstract algebra and number theory. It introduces an array of fundamental structures and shows how to balance intuition and rigor.


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 rigorand the flexible thinkingrequired 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 springer.com forinstructors adopting the text for a course.

Solutions manual for even numbered exercises is available on springer.com 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 Request lecturer material: sn.pub/lecturer-material

Autorentext
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.

Inhalt
Preface to the Second Edition.- Preface to the First Edition.- 1. Logic.- 2. Sets.- 3. Functions.- 4. Finite and Infinite Sets.- 5. Combinatorics.- 6. Number Theory.- 7. Complex Numbers.- Hints and Partial Solutions to Selected Odd-Numbered Exercises.- Index.

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