The authors have given themselves a difficult task: write a new discrete mathematics textbook, including such topics as abstract algebra and linear algebra. This could be a welcome addition to the current crop of texts. Unfortunately, the book contains many errors--in English usage, mathematics, and production--that severely limit its usefulness.
The text contains 11 chapters: “Sets, Relations and Functions,” “Combinatorics,” “Mathematical Logic,” “Algebraic Structures,” “Matrix Algebra,” “Order, Relation [sic] and Lattices,” “Boolean Algebra,” “Complexity,” “Graph Theory,” “Tree [sic],” and “Formal Language [sic] and Automata.” As evident in the chapter headings, the authors have a cavalier attitude toward plurals and articles. Here are other examples:
- “Growth of functions is intended to compare relative sizes of functions that is very useful in the analysis of computer algorithms” (p. 61).
- “Gaussian elimination method is an efficient numerical method and can be implemented on high-speed digital computer” (p. 266).
One problem with the poor English is that at times it confuses the mathematics. Take, for example, this definition of a cyclic group (p. 185): “A group is called cyclic if for some a in G, there exists element x in G, of the form an where n is some integer.” This is wrong as it is written; maybe the authors are confusing “there exists” with “for all”?
There are similar examples on almost every page. However, most of the mathematics is at least correct; there are also a number of errors, of which these are but a sample:
- A lattice is defined as being a poset in which any two elements have both a supremum and an infimum (p. 296). However, these must be unique or else Figure 6.28 (i) would be a lattice.
- Example 5.18 (p. 255) seems to indicate that the rank of a matrix is equal to the number of nonzero rows, after some elementary row operations. This is false. The rank is the number of nonzero rows in the echelon form of the matrix (and this definition is in fact provided a few pages further on). This is a better definition than requiring determinants.
- The authors state (p. 538): “Any problem in P-class is also in NP-class (since a P-class problem can be solved non-deterministically) but the reverse is not true.” Does this mean that the authors believe that polynomial time (P) = nondeterministic polynomial time (NP) is false?
One major problem with this book is that there is simply too much in it. Too often a topic is introduced, briefly described, and then discarded (examples of such topics are determinants and eigenvalues and abstract algebra). The authors missed an opportunity to discuss matrix groups in one chapter and rings and fields in the next. The book would be greatly improved if there were fewer topics and more connections between them. This would provide greater homogeneity; it is also sound pedagogy.
There are some curious omissions. Although permutation groups are discussed, there is no mention of Cayley’s theorem--surely a superb reason for introducing permutation groups at all in an introductory text. There is no number theory except as an appendix; this again is a missed opportunity, for then the authors could have discussed finite fields and even provided some simple cryptographic examples. There is no discussion of algorithmics, nor is there any use of modern mathematical software. There are some--though not nearly enough--examples of pseudocode scattered throughout the text, but the authors should have used either a programming language (Python, for example) or an open-source computer algebra system. Well ordering is introduced, but without any discussion of the well-ordering principle. Again, a topic is briefly mentioned without any connection to previous material--the book is full of examples.
There are also some topics that would have benefited from a more detailed discussion. Methods of proof, for example, get a mere seven pages--such a vital topic deserves its own chapter. NP-complete problems, such as graph isomorphism and finding Hamiltonian paths, are also given far less space than they deserve.
The diagrams have a tendency to be sloppy: graph edges meet vertices in random places, and vertices have different sizes from diagram to diagram, appear as ellipses in several cases, and in at least one figure (Exercise 9, p. 435) double up. Given the existence of high-quality, open-source mathematical diagram software, there is no excuse for anything less than excellence.
I also take exception with some of the text’s pedagogy. Each chapter begins with some “Learning Objectives”--a very important addition to any text--but these all claim that after “reading” the chapter “you will be able to understand the following.” The term “understand” is simply too vague and meaningless to be used as a learning objective. Finally, the index is a bit hit or miss. Some topics don’t exist (for example, echelon form and mathematical induction) and other entries (for example, “Cauchy’s”) are confusing.
In conclusion, this text has the potential to be a useful addition to the market. However, in its current state, I cannot fully recommend it. It needs to be significantly tidied up and released in a corrected second edition before it can be widely used.
