Discrete mathematics is often characterized as the totality of those branches of mathematics studying the various structural properties of countable sets. In particular, the term “finite mathematics” refers to those parts of discrete mathematics that are devoted to finite sets and their diverse properties. Among the major topics in discrete mathematics are such central areas as general logic, computability and recursion theory, set theory, proof theory, algebraic structures, combinatorics, graphs and trees, the theory of computation, formal languages, the theory of abstract automata, and others. In the second half of the 20th century, discrete mathematics has become a basic constituent of both computer science and information theory, which prompted a tremendous increase of research in this extremely versatile, interdisciplinary field.
The book under review provides a first introduction to several foundational aspects of discrete mathematics. As such it may serve as solid textbook for undergraduate and postgraduate students in mathematics, computer science, and information technology likewise.
As for the contents, this book contains ten chapters, each of which is subdivided into several sections.
Chapter 1 introduces the principle of mathematical induction, the well-ordering principle, and some elementary number theory, including prime numbers and linear congruences. Chapter 2 treats the rudiments of naive set theory, together with a little combinatorics and counting formulas. Chapter 3 is devoted to some aspects of classical mathematical logic, ranging from the principles of logical reasoning to the first steps into predicate logic. Chapter 4 deals with general functions, relations, permutations, various types of recursive functions, and several concrete examples. Chapter 5 explains the basics of group theory, along with very brief introductions to monoids, rings and fields, respectively. Chapter 6 turns to Hamming codes, while Chapter 7 is concerned with posets, Hasse diagrams, lattices, and Boolean algebras. Chapter 8 discusses recurrence relations and generating functions, mainly through concrete examples. Chapter 9 presents the basics of the theory of graphs and trees in a concise way, again with a strong focus on examples and solved problems. Finally, Chapter 10 gives an introduction to the principles of the theory of computation, with a special focus on formal languages, phase-structure grammars, context-free grammars, deterministic and non-deterministic abstract automata, push-down automata, regular sets as languages accepted by finite automata, finite-state automata, and Turing machines.
In fact, the author has set high value on explaining the abstract concepts and results by an incredibly large number of concrete examples, solved problems, and related exercises. It seems fair to state that the whole text is largely example-driven, which is certainly a key feature of this very instructive textbook. Also, the material is presented in an utmost detailed and user-friendly style of writing, which makes the book a useful primer for beginners in the field of discrete mathematics.