This book is designed to be used in a one- or two-semester course in discrete structures in mathematics or computer science departments. A tree diagram is given to describe possible collections of chapters so that it is clear which chapters are prerequisites of others.

The first chapter introduces graphs and a “hard” Hamiltonian circuit problem for graphs as a recreational activity.  The note at the end of the chapter briefly mentions NP-complete problems but says little about them.  Graphs are taken up again in the second-to-last chapter on graphs, Chapter 14.  This chapter is perhaps one of the most important for computer science majors.  The subsection on trees is especially important yet given short shrift, receiving even less attention than the coloring of graphs. Many less-experienced teachers might not include the chapter at all when teaching a discrete structures course. 

Chapter 2 focuses on the fundamentals of logic and logical arguments, ending with a section on formal proofs. Once again, the authors refer to algorithmic complexity with no indication of what it all means. The notes at the end of chapters are interesting and put the topics in historical perspective, but they should give complete definitions in the main text.

Chapter 3 on sets, relations, functions, and the cardinality of sets is important for all of the remaining chapters, as is Chapter 4 on proofs by induction.  It is rare to see the notion of well-founded induction discussed in discrete mathematics books, and the authors explain it well. The same can be said for structural induction, particularly as it relates to recursively-defined functions.  In the notes the authors indicate that various methods of induction are used in computer science for program verification.  Why not give an example?

Chapters 5 and 6 on equivalent relations, partial orders, lattices, and the use of Hasse diagrams cover fundamental topics for computer science students to understand and use.  If I were teaching the course, I would prefer more exercises, especially some that directly apply to computer science.  The same can be said for the floor and ceiling functions of Chapter 7. Why are they included here? Chapter 8 contains some number theory and the RSA public key cryptosystem.  Why not include some elementary cryptographic tools and a little of their history?

Chapter 9 on sums includes material on difference calculus with an example analyzing a quicksort algorithm’s run time, a nice specific example of an application useful in computer science. Similarly, Chapter 10 compares functions to give an asymptotic approximation of one function by another to ultimately determine a function’s order of growth. 

Most teachers would have reached the end of the semester at this point, thus leaving some important topics out or for a second-semester course.  In particular, basic counting techniques are studied in Chapter 11 and continued into Chapter 12 on generating functions.  Including more examples drawn from computer science (at least in the exercises) would be useful.  Tucker’s Applied Combinatorics provides excellent examples of such applications.  Chapter 13 continues the theme of counting problems by introducing recurrence relations and Catalan numbers, both of which have applications in computer science.

As mentioned in the beginning, Chapter 14 is a chapter on graphs, yet as important as graphs (and especially trees) are in many areas of computer science, little attention is given to this applicability.  The same can be said for the concluding Chapter 15 on probability.

I take issue with the second sentence in the preface that says that discrete mathematics is not a mathematical discipline itself but a “mélange of topics from logic, set theory, algebra, combinatorics, number theory, and other areas of mathematics.”  I work in discrete mathematics and have written an extensive number of research papers in applied discrete mathematics.  Discrete mathematics has subareas, just as abstract algebra has numerous subareas. The authors say they are providing coherence to seemingly incongruent subjects.  I say they are doing quite the opposite.  The fifteen chapters are a mixture of topics, none of which are structurally linked together.  Since the authors rarely give clear applications of these topics to computer science, their importance in computer science is barely indicated.

In summary, I have a hard time recommending this book for undergraduate classes in either discrete mathematics or discrete structures classes at the early undergraduate level.  The bibliography is extensive for the more theoretical topics but does not contain classic books in applied graph theory and combinatorics such as those by Roberts and Tucker.

Margaret (Midge) Cozzens is a Distinguished Research Professor at Rutgers University and Associate Director for Education at the DIMACS Center (Discrete Mathematics and Theoretical Computer Science Center).  She taught discrete mathematics and discrete structures courses at Northeastern University and the University of Colorado Denver in the early 2000s, and she has run many workshops for faculty in discrete mathematics.  She has had a number of REU and graduate students work on problems and projects related to applications of discrete mathematics.