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Principles of Mathematics: A Primer

Vladimir Lepetic
John Wiley
Publication Date: 
Number of Pages: 
[Reviewed by
Miklós Bóna
, on

Books that introduce undergraduate students to higher mathematics are numerous. One common complaint against a significant portion of them is that they talk a lot about how to prove theorems without actually proving anything interesting. It is as if a dinner host discussed the details of eating a nice meal, but only served chips and water.

Some authors avoid that trap by choosing one part of higher mathematics and using that part to show the students the power of their new theorem-proving tools. The book under review goes down this path, choosing group theory and linear algebra to be the areas in which substantial results will be proved in Chapters 5 and 6 (the last two chapters in the book). The author goes deep enough into group theory to cover normal subgroups and isomorphism theorems, and deep enough into linear algebra to discuss eigenvalue theory and the general linear group.

A few choices that the author makes in ordering his early chapters are rather surprising. Chapter 3, which starts on page 183, is called Proofs, but by then we have done many proofs which were called just that. Functions are the topic of Chapter 4, but Set Theory is Chapter 1, and in that first chapter, we prove, for instance, that the rationals are equinumerous to the integers, using bijective functions of course. We also prove that there is no bijection between the reals and the integers.

These choices can perhaps be explained in a somewhat contrived way, for instance by saying that students already have an idea of what a function is, what proofs are, and we will formalize these concepts later, but this reviewer thinks that a book that intends to introduce readers into the world of proof-based mathematics is not the place for this kind of non-linear coverage.

There are plenty of exercises and supplementary problems, though none of them come with solutions. Many sentences that end in a math formula do not have a period at the end, which sometimes makes reading harder than it should be. To summarize, the book is certainly different from the competition, but more editing, and especially a more straightforward ordering of the topics, would have improved it.

Miklós Bóna is Professor of Mathematics at the University of Florida.

Preface xi

1 Set Theory 1

1.1 Introduction, 1

1.2 Set Theory – Definitions, Notation, and Terminology – What is a Set?, 3

1.3 Sets Given by a Defining Property, 15

1.4 The Algebra of Sets, 25

1.5 The Power Set, 41

1.6 The Cartesian Product, 44

1.7 The Sets N, Z, and Q, 46

1.8 The Set R – Real Numbers I, 71

1.9 A Short Musing on Transfinite Arithmetic, 80

1.10 The Set R – Real Numbers II, 102

1.11 Supplementary Problems, 109

2 Logic 115

2.1 Introduction, 116

2.2 Propositional Calculus, 121

2.3 Arguments I, 146

2.4 Arguments II, 167

2.5 A Short Revisit to Set Theory, 171

2.6 Boolean Algebra, 173

2.7 Supplementary Problems, 177

3 Proofs 183

3.1 Introduction, 183

3.2 Direct Proof, 193

3.3 Indirect Proof, 212

3.4 Mathematical Induction, 218

3.5 Supplementary Problems, 241

4 Functions 247

4.1 Introduction, 247

4.2 Relations, 248

4.3 Functions, 274

4.4 Supplementary Problems, 321

5 Group Theory 327

5.1 Introduction, 327

5.2 Fundamental Concepts of Group Theory, 328

5.3 Subgroups, 356

5.4 Cyclic Groups, 382

5.5 Homomorphisms and Isomorphisms, 385

5.6 Normal Subgroups, 404

5.7 Centralizer, Normalizer, Stabilizer, 412

5.8 Quotient Group, 419

5.9 The Isomorphism Theorems, 427

5.10 Direct Product of Groups, 437

5.11 Supplementary Problems, 441

6 Linear Algebra 447

6.1 Introduction, 447

6.2 Vector Space, 449

6.3 Linear Dependence and Independence, 456

6.4 Basis and Dimension of a Vector Space, 461

6.5 Subspaces, 469

6.6 Linear Transformations – Linear Operators, 477

6.7 Isomorphism of Linear Spaces, 489

6.8 Linear Transformations and Matrices, 501

6.9 Linear Space Mmn, 507

6.10 Matrix Multiplication, 509

6.11 Some More Special Matrices. General Linear Group, 514

6.12 Rank of a Matrix, 525

6.13 Determinants, 534

6.14 The Inverse and the Rank of a Matrix Revisited, 541

6.15 More on Linear Operators, 547

6.16 Systems of Linear Equations I, 585

6.17 Systems of Linear Equations II, 600

6.18 The Basics of Eigenvalue and Eigenvector Theory, 613

6.19 Supplementary Problems, 635

Index 645