Discrete invitations

A peculiar feature of books on discrete mathematics is that their authors feel the need to define what discrete mathematics is. These textbooks are no different: Stephen Barnett describes it as the area constituted by "the principles and applications of manipulating and processing numbers", Jiri Matousek and Jaroslav Nesetril as the mathematics of objects which "are clearly separated and distinguishable" and mathematics "dealing with finite sets".

Barnett's definition is unusual in its emphasis on numbers. In Discrete Mathematics: Numbers and Beyond , this is reflected in the topics, with chapters on number theory and number systems, combinatorics, codes, recurrence relations, and graphs. The omission of algorithms and the inclusion of coding theory are somewhat surprising. Barnett explains that the book is aimed at first-year undergraduates in disciplines such as finance, economics and engineering, but not computer science. But it seems that a course taught from this book, combined with a course on algorithms and formal logic, would form a good foundation for the theoretical part of a computer-science degree.

The book is definitely aimed at the non-mathematical student. The word "proof" is hardly used, and the word "theorem" does not appear until page 84. Theory is explored by looking at specific examples, after which more general theory is developed.

There are about 750 exercises, from routine questions to "student projects", and the book has an attractive layout.

Not only is the definition of discrete mathematics in Invitation to Discrete Mathematics more classical, the rest of the book suggests it is more old-fashioned. Topics include counting, generating functions, graph theory, projective geometry, probabilistic arguments, and applications of linear algebra. The text is built around definitions, theorems and proofs. And the layout, based on rather dull LaTeX styles and fonts, completes the picture of a traditional textbook. But when one starts reading, it emerges as a far-from-traditional textbook and is a joy to read. The text is lucid and sprinkled with small jokes and background stories. Footnotes, which in many texts constitute the most boring parts, here contain some of the best: a footnote on page 169 explains why "doughnut" is not politically correct and hence not used to describe a torus.

The level of expertise means the book is aimed at a more advanced undergraduate or beginning-graduate course. The manner of presentation makes it primarily suitable for students in mathematics, in particular those who can appreciate the mathematics for its own sake, the extra reading material offered and the free writing style. There are about 450 exercises, again ranging from the straightforward to the fairly complex.

Jan van den Heuvel is lecturer in mathematics, London School of Economics.