Béla Bajnok - An Invitation to Abstract Mathematics

It has been more than three decades since the first so-called transition book appeared on the mathematics shelves of college bookstores, and there are currently several dozen such books available. The aim of these books is to bridge the gap between the traditional lower-level courses, primarily calculus, and the upper-level courses that require deeper understanding and maturity, such as modern algebra and real analysis. Thus, the main focus of transition books is on the foundations of abstract mathematics, giving a thorough treatment of elementary logic and set theory and introducing students to the art and craft of proof writing.

While this book certainly hopes to provide students with a firm foundation for the upper-level courses of an undergraduate mathematics program, it is not geared solely toward students who intend to major in mathematics. It is the disappointing reality at many institutions that some of the most able students are not considering mathematics as a possible major; in fact, coming out of a standard calculus sequence, most students are not familiar with the true nature of this beautiful subject. Therefore, an important mission of the book is to provide students with an understanding and an appreciation of (abstract) mathematics, with the hope that they choose to study these topics further.

Recognizing that not all our students will have the opportunity to take additional courses in mathematics, this textbook attempts to give a broad view of the field. Even students majoring in mathematics used to complain that they were not given an opportunity to take a course on mathematics without an artificial division of subjects. In this textbook, we make an attempt to remedy these concerns by providing a unified approach to a diverse collection of topics, by revisiting concepts and questions repeatedly from differing viewpoints, and by pointing out connections, similarities, and differences among subjects whenever possible. If, during or after reading this book, students choose to take further courses in mathematics, then we have achieved our most important goal.

In order to provide students with a broad exposure to mathematics, we have included an unusually diverse array of topics. Beyond a thorough study of concepts that are expected to be found in similar books, we briefly discuss important milestones in the history of mathematics and feature some of the most interesting recent accomplishments in the field. This book aims to show students that mathematics is a vibrant and dynamic human enterprise by including historical perspectives and notes on the giants of mathematics and their achievements; by mentioning more recent results and updates on a variety of questions of current activity in the mathematical community; and by discussing many famous and less well-known questions that have not yet been resolved and that remain open for the mathematicians of the future.