Probably my first observation of this book was that, like many Springer books, it’s hardcore. I do not mean that the material is particularly difficult or intractable, but that the book does not pander to the reader. There is no breezy humor, nor are there long explanations. Rather, it gets straight down to the business of the topics. The topics, as the first part of the name suggests, are logic and mathematics. While there are some computer science topics, I feel a more accurate title may have been Logic and mathematics for computer science. This is very much a book that teaches the mathematics that underlies a lot of computer science (also often referred to as discrete mathematics).

After some brief introductory material, the book jumps straight into propositional logic. In under 100 pages, it covers a lot of material. While this book is probably suitable for self-guided study, and probably can be understood by someone with only high school mathematics, it would take a lot of discipline and focus to work through it. On the other hand, it is quite suitable for a first- or second-year university course where the hand of the lecturer can guide the student through the material. There are fairly comprehensive sets of problems at the end of each chapter, and solutions are given to many of the odd numbered problems. The same observation about the level of focus required applies to the rest of the book.

Predicate logic and first-order logic occupy about the first third of the book. The next part introduces set theory, which also runs into mathematical functions. Once again, the material is solid and comprehensive for the level at which it is aimed. This material also covers roughly one-third of the book.

The third part of the book (note that the distinctions between the three different parts of the book are mine, not imposed by the book) builds on set theory and introduces the integers and the rationals. Chapter 4, “Mathematical Induction,” lays the groundwork for the integers and rationals. Since I have a computer science background, it is hard to assess what it would be like for someone without this background to read the later chapters and assimilate them, but I suspect it would be rather challenging. The material is dense and unforgiving, but thorough. Those who prefer definitions, axioms, and theorems to prose will find the approach welcoming.

The penultimate chapters delve more deeply into set theory (chapter 5: “Well-Formed Sets: Proof by Transfinite Induction with Already Well-Ordered Sets” and chapter 6: “The Axiom of Choice: Proofs by Transfinite Induction”). Finally, the last chapter is perhaps the “chattiest” of the chapters and looks at the applications of earlier work. Chapter 7, “Applications: Nobel-Prize Winning Applications of Sets, Functions, and Relations,” looks at some practical applications of the work introduced in the earlier sections.

In summary, the book is a very thorough treatment of the topics mentioned above. It is quite suitable as a textbook for an undergraduate course in the logic and mathematics that underpin a computer science course. The exercises are comprehensive. I would probably not recommend this book to someone without a solid mathematical background as an introduction to the topics for self-guided learning.