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An Invitation to Abstract Mathematics

Béla Bajnok
Publication Date: 
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Undergraduate Texts in Mathematics
BLL Rating: 

The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Jill Dietz
, on

There are dozens of textbooks designed to help mathematics majors “transition” to higher-level mathematics and mathematical thinking, but Bajnok’s new book truly invites students to enjoy the beauty, power, and challenge of abstract mathematics.

The book can be used as a text for traditional transition or structure courses — indeed, this is the most likely market for it — but since Bajnok invites all students, not just mathematics majors, to enjoy the subject, he assumes very little background knowledge.

In the Introduction to Abstract Mathematics course that Bajnok teaches at Gettysburg College, students come with a wide variety of backgrounds. Some are math enthusiasts who may not have taken a calculus course and may never do so, while others may have taken a couple of years of college-level mathematics. To put them all on the same playing field, the book and the course begin with a set of games that are easy to play and understand, yet allow the author to eventually explore fundamental notions in mathematics such as precise language, truth, proof and conjecture, abstract structures, and more.

What is in the book?

The book is divided into three parts. Part I (What’s Mathematics?) starts a “friendship with abstract mathematics” by introducing readers to a variety of games that will be used to illustrate multiple concepts and methodologies throughout the book. Bajnok begins with a Hackenbush game that he calls Aerion. It is easy to play and therefore easily understood by anyone with an affinity for logic, but has enough complexity that it can be called upon over and over again to provide a setting for understanding the need for clear definitions and statements and for developing the idea of truth in mathematics. These notions together with some classic theorems (e.g., \(\sqrt{2}\) is irrational) and famous recent theorems (e.g., the four color theorem) round out the first part.

Part II (How to Solve It?) is where one finds the more traditional transition topics. Anyone with knowledge of this genre of books will recognize the usual chapters on logic, sets, quantifiers, and methods of proof including induction, existential and universal proofs.

Bajnok’s book is different from the rest in a couple of ways. First, his selection of examples and problems covers the usual (e.g., some famous sums in the induction chapter) as well as the unusual. For example, a problem on “perfect square dissection” that is more visual in nature than standard summation problems or Pascal’s Triangle identities is in the induction section; furthermore, as with many exercises, Bajnok uses the opportunity to give readers some history of the problem and its solution.

Second, Bajnok uses logic and set theory as motivation for studying what turns out to be group, ring, field, and Boolean algebra structures. In the set theory chapter (Setting Examples) he asks one to notice the similarity between the algebra of statements and the algebra of sets. For example, one can see the similarity between the logical tautology \[ (P \vee \neg Q \vee R) \wedge (\neg (P\vee R)) \Leftrightarrow \neg P \wedge \neg Q \wedge \neg R\] and the set identity \[ (A \cup \overline{B} \cup C) \cap \overline {(A\cap C)} = \overline{A} \cap \overline{B} \cap \overline{C}.\]

Abstracting properties of statements and sets, one arrives at the usual axioms needed to build a variety of algebraic structures. This is a dense section of the book, but the formal axiomatic proofs will benefit students who intend to continue their study of mathematics in courses such as abstract algebra and real analysis.

Bajnok covers the usual introductory group, ring and field examples — the integers, various number fields, the symmetric group — and some innovative ones. In keeping with his invitation to a wide variety of readers to enjoy abstract mathematics, Bajnok uses games to illustrate algebraic properties. For example, associated to the game of Nim are Nim addition and Nim multiplication. Justification for the definitions lies in the fact that a winning strategy is associated to a Nim sum being zero. Other games utilizing Nim addition and multiplication are Acrostic Twins and Turning Corners. A winning strategy for Turning Corners is associated to a Nim product being zero. The fascinating punchline is that Nim addition and multiplication on the set {0, 1, 2, 3} define a field of order 4. The existence of such a field often requires a lot of build-up about polynomial quotient rings in a course and textbook dedicated to abstract algebra topics, but Bajnok does it quite naturally by playing games.

Part III (Advanced Math for Beginners) has chapters on relations, functions, limits, counting, number systems, and more combinatorial game theory. Again, Bajnok covers some topics that are found in many transition texts that will help students prepare for full courses in real analysis and abstract algebra, as well as unusual topics. Chapter 23 builds up number systems axiomatically, beginning with using set theory to build the natural numbers. The chapter is highly abstract, but shows the power of the theory Bajnok so carefully developed in the previous chapters. In Games are Valuable, Bajnok returns to the combinatorial game theory topics that began the book, but with more rigor and attention to detail.

If a reader has made it this far, he or she will have a substantial grounding in the essence of mathematical theory and will have been exposed to both fundamental mathematical notions and modern day topics.

Who is the book for?

Bajnok’s book can most certainly be used as a text for a traditional transition course designed for mathematics majors. He wrote the book with a broader audience in mind, and personally uses it at Gettysburg College for anyone interested in deepening his or her mathematical knowledge, regardless of major.

Bajnok notes that his Hungarian roots are revealed by the book’s focus on problem-solving. Indeed, it would be difficult to deliver a 50-minute lecture from a chapter of the book, but it is designed perfectly for anyone who runs an active-learning classroom. The chapters in Part I can be read by a student on his or her own, so that class time can be used for solving problems and presenting their solutions. Part II might take a little more intervention on the professor’s part, presenting some of the more difficult topics in class before settling down to problem-solving. Bajnok, by the way, has his students read a section and prepare solutions to assigned problems outside of class. In “exploratorium” sessions, also outside of class, students work on problems under the supervision of a teaching assistant so that class time is spent almost entirely on presentations of problems. This seems like a ``Hungarian modified Moore method” that works incredibly well for Bajnok, and could be used by others who are comfortable with this kind of “flipped” classroom.

While many of Bajnok’s topics are familiar to those who know the transition genre, this is not a run-of-the-mill textbook that you can get away with reading just a few hours before walking into your classroom. One would want to take some time getting to know the content and especially the exercises before using the book. As Bajnok writes in his Preface to Instructors, “the lectures and the problems build on one another; the concepts of the lectures are often introduced by problems in previous chapters or are extended and discussed again in problems in subsequent chapters.”

I have never used the book myself, but can imagine that most of the first two parts along with a favorite section or two from Part III would make a nice introduction to abstract mathematics.

Price and other matters

Springer lists the price of Bajnok’s book in hardcover at $59.95, though it is easy to find it for under $50. This is quite affordable for something with such incredible depth and breadth of content. Furthermore, it is a Springer e-book so a digital version is quite a bit cheaper, or even free if one’s college or university library offers free access to SpringerLink to their students and faculty.

As Bajnok notes “there are very few routine problems; most problems will require relatively extensive arguments, creative approaches, or both.” For this reason, it is especially good news that an instructor’s guide with problem solutions and further discussions is available from the publisher.

Jill Dietz is Professor of Mathematics and chair of the Department of Mathematics, Statistics, and Computer Science at St. Olaf College in Northfield, Minnesota.

Preface to Instructors
Preface to Students

I What's Mathematics
1 Let's Play a Game!
2 What's the Name of the Game?
3 How to Make a Statement?
4 What's True in Mathematics?
5 Famous Classical Theorems
6 Recent Progress in Mathematics

II How to Solve It?
7 Let's be Logical!
8 Setting Examples
9 Quantifier Mechanics
10 Mathematical Structures
11 Working in the Fields (and Other Structures)
12 Universal Proofs
13 The Domino Effect
14 More Domino Games
15 Existential Proofs
16 A Cornucopia of Famous Problems

III Advanced Math for Beginners
17 Good Relations
18 Order, Please!
19 Let's be Functional!
20 Now That's the Limit!
21 Sizing It Up
22 Infinite Delights
23 Number Systems Systematically
24 Games Are Valuable!

IV. Appendices
A. Famous Conjectures in Mathematics
B The Foundations of Set Theory
C All Games Considered
D Top 40 List of Math Theorems