This book is designed as a first one-semester course in discrete mathematics for mathematics and computer science students.  As a computer science course it would be called a discrete structures course. 

The first and second chapters provide a basic introduction to the natural numbers, mappings, and the principles of enumeration, including maps, permutations and combinations, binomial and multinomial coefficients, lattices and Catalan numbers along with various combinatorial identities. Some of the calculations in computing the truth of the identities are at best confusing to a novice reader, yet the exercises require a thorough understanding of these proofs.  The third chapter focuses on the principle of inclusion-exclusion and its applications. The applications are not what many would call applications, but are rather number theoretical applications of earlier propositions, using not just the principle of inclusion-exclusion but often also earlier results.

Chapter 4 starts with the easy-to-understand pigeonhole principle.  Teachers and students will appreciate the description of the pigeonhole principle that begins Section 4.1.  Among the provided examples, I would have ordered Example 4.1.3 first since it is more intuitive and involves the number of acquaintances a person has in a meeting. The chapter continues by focusing on partial orders, ultimately proving Sperner’s Theorem and the Erdős-Ko-Rado Theorem. Generalizations of these theorems are reserved for the exercises. As in most chapters, the exercises in general are apt to be hard for even the best students.

Chapter 5 on graphs provides an introduction to many aspects of graph theory, including trees, colorings, matchings, and planar graphs.  As a graph theorist, I would have appreciated some applications; however, all of the applications are reserved for the exercises, and few are included.

Chapter 6 concerns generating functions, and is a rigorous treatment of the topic, probably challenging for almost all first-year students.  Fitting this chapter and Chapter 7 on discrete probability into a one-semester discrete mathematics course for first- and second-year students will be difficult, assuming an appropriate amount of time is spent on the first five chapters.

Throughout the book, the exposition is straightforward with sufficient detail for complete understanding. As the preceding summary of topics covered in this book makes clear, a considerable amount of material is covered in a relatively small space. Each section ends with an assortment of exercises (between 30 and 60 per section), many of which are quite difficult.

I have taught for the better part of fifty years, and it seems to me that the topics covered here are not optimal for a one-semester first course in discrete mathematics, especially to first- and second-year undergraduates, where most discrete mathematics and discrete structures courses occur in the curriculum. Athanasiadis’s book is instead more suitable for strong students, and contains interesting problems, some better assigned as projects.  In difficulty it is perhaps comparable to Stanley’s Enumerative Combinatorics graduate text. Written at a freshman or sophomore level is Ken Rosen’s discrete mathematics book, now in its 8^th edition, a tribute to its usability.  It takes students where they are and focuses on first- and second-year students. Ken Bogart’s book in discrete mathematics is also very usable—unfortunately Ken Bogart was killed in a cycling accident, so new versions are unlikely.

Margaret (Midge) Cozzens is a Distinguished Research Professor at Rutgers University and Associate Director for Education of the DIMACS Center (The Center for Discrete Mathematics and Theoretical Computer Science). She has taught undergraduate and graduate discrete mathematics at Northeastern University, the University of Colorado at Denver, and at Rutgers.