An Invitation to Abstract Mathematics

✍ Scribed by Béla Bajnok (auth.)

• Publisher: Springer-Verlag New York
• Year: 2013
• Tongue: English
• Leaves: 406
• Series: Undergraduate Texts in Mathematics
• Edition: 1
• Category: Library

✦ Synopsis

This undergraduate textbook is intended primarily for a transition course into higher mathematics, although it is written with a broader audience in mind. The heart and soul of this book is problem solving, where each problem is carefully chosen to clarify a concept, demonstrate a technique, or to enthuse. The exercises require relatively extensive arguments, creative approaches, or both, thus providing motivation for the reader. With a unified approach to a diverse collection of topics, this text points out connections, similarities, and differences among subjects whenever possible. This book shows students that mathematics is a vibrant and dynamic human enterprise by including historical perspectives and notes on the giants of mathematics, by mentioning current activity in the mathematical community, and by discussing many famous and less well-known questions that remain open for future mathematicians.

Ideally, this text should be used for a two semester course, where the first course has no prerequisites and the second is a more challenging course for math majors; yet, the flexible structure of the book allows it to be used in a variety of settings, including as a source of various independent-study and research projects.

✦ Table of Contents

Front Matter....Pages i-xiv
Front Matter....Pages 1-1
Let’s Play a Game!....Pages 3-10
What’s the Name of the Game?....Pages 11-22
How to Make a Statement?....Pages 23-31
What’s True in Mathematics?....Pages 33-44
Famous Classical Theorems....Pages 45-54
Recent Progress in Mathematics....Pages 55-68
Front Matter....Pages 69-69
Let’s Be Logical!....Pages 71-81
Setting Examples....Pages 83-94
Quantifier Mechanics....Pages 95-108
Mathematical Structures....Pages 109-123
Working in the Fields (and Other Structures)....Pages 125-133
Universal Proofs....Pages 135-147
The Domino Effect....Pages 149-160
More Domino Games....Pages 161-173
Existential Proofs....Pages 175-187
A Cornucopia of Famous Problems....Pages 189-195
Front Matter....Pages 197-197
Good Relations....Pages 199-212
Order, Please!....Pages 213-227
Let’s Be Functional!....Pages 229-239
Now That’s the Limit!....Pages 241-257
Front Matter....Pages 197-197
Sizing It Up....Pages 259-281
Infinite Delights....Pages 283-310
Number Systems Systematically....Pages 311-342
Games Are Valuable!....Pages 343-380
Back Matter....Pages 381-406

✦ Subjects

Mathematics, general; History of Mathematical Sciences; Mathematical Logic and Foundations