A Transition to Advanced Mathematics (4th Edition)
Quick Informations
Where to buy it: Amazon
Authors: Douglas Smith, Maurice Eggen and Richard St. Andre
Release date: August 15, 1997
Language: English
Personal Progress: Read completely
Review
Description
A Transition to Advanced Mathematics is the perfect book to get into pure mathematics and rigorous proofs after taking the usual calculus classes. The book is structured in a way that gives a very large and clear overview of what mathematics really are. Moreover, since there is a big focus on proofs, every single theorem can be proved using what was already proved in the book, in other words, the book is completely self-contained. I discovered in this book the satisfaction of proving every single steps leading to a theorem.
Prerequisites
This is an introduction to Pure Mathematics, the only prerequisites would be to be familiar with highschool mathematics : basic algebra and trigonometry. Some examples and exercises use the concepts of derivatives and integrals so it would be preferable to have a background in calculus. Since the last chapter is on Analysis, then again, a background in calculus would really help.
Content
I find the content really interesting. The book starts with a chapter on Logic which was surprising but turned out to be extremely important. I was amazed by the power of the tools presented in this chapter. The second chapter is on Set Theory (more precisely, what we now call Naive Set Theory). The notion of axioms and natural numbers are discussed. The third chapter discusses the notion of relations, equivalence relations and orderings. Before this chapter, I had strictly no idea what a relation was and I was afraid that the chapter would be boring. The chapter turned out to be really interesting as well as all of the other chapters. The transition from Chapter 3 to Chapter 4 was very smooth considering functions as a special case of relations. Chapter 5 explores rigorously the notion of cardinality, I personaly found this chapter to be the hardest, but still, an amazing chapter with very deep content (from the point of view of a complete beginner in pure mathematics). Finally, the books perfectly ends with a sixth chapter on Group Theory (which opens the doors of abstract algebra) and a seventh chapter on Analysis (which opens the doors of Real Analysis).
Exercises
This book has a lot of exercises and most of them really require to sit and think as hard as you can. This was my first time spending this much time on a single exercise, and I think that completing all of them really prepared me for my first pure mathematics classes. If I have time, I should probably write my solutions in LaTeX and put them in the Textbook Solutions page of this website.
Further Readings
Since the last two chapters open the doors of Abstract Algebra and Analysis, then this book gives you the perfect prerequisites to start an introductory book on Abstract Algebra or Analysis. I would say that the best follow-up book on Abstract Algebra would be Abstract Algebra by Dummit & Foote. In Analysis, a book that seems to be the perfect sequel to this one would be Understanding Analysis by Stephen Abbott.