In a comprehensive yet easy-to-follow manner, Discrete Mathematics for New Technology follows the progression from the basic mathematical concepts covered by the GCSE in the UK and by high-school algebra in the USA to the more sophisticated mathematical concepts examined in the latter stages of the book. The book punctuates the rigorous treatment of theory with frequent uses of pertinent examples and exercises, enabling readers to achieve a feel for the subject at hand. The exercise hints and solutions are provided at the end of the book. Topics covered include logic and the nature of mathematical proof, set theory, relations and functions, matrices and systems of linear equations, algebraic structures, Boolean algebras, and a thorough treatise on graph theory.
Although aimed primarily at computer science students, the structured development of the mathematics enables this text to be used by undergraduate mathematicians, scientists, and others who require an understanding of discrete mathematics.
Table of Contents
Logic: Propositions and truth values. Logical connectives and truth tables. Tautologies and contradictions. Logical equivalence and logical implication. The algebra of propositions. More about conditionals. Arguments. Predicate logic. Arguments in predicate logic. Mathematical proof: The nature of proof. Axioms and axiom systems. Methods of proof. Mathematical induction. Sets: Sets and membership. Subsets. Operations on sets. Counting techniques. The algebra of sets. Families of sets. The cartesian product. Relations: Relations and their representations. Properties of relations. Intersections and unions of relations. Equivalence relations and partitions. Order relations. Hasse diagrams. Application: relational databases. Functions: Definitions and examples. Composite functions. Injections and surjections. Bijections and inverse functions. More on cardinality. Databases: functional dependence and normal forms. Matrix algebra: Introduction. Some special matrices. Operations on matrices. Elementary matrices. The inverse of a matrix. Systems of linear equations: Introduction. Matrix inverse method. Gauss-Jordan elimination. Gaussian elimination. Algebraic structures: Binary operations and their properties. Algebraic structures. More about groups. Some families of groups. Substructures. Morphisms. Group codes. Boolean algebra: Introduction. Properties of Boolean algebras. Boolean functions. Switching circuits. Logic networks. Minimization of Boolean expressions. Graph theory: Definitions and examples. Paths and circuits. Isomorphism of graphs. Trees. Planar graphs. Directed graphs. Applications of graph theory: Introduction. Rooted trees. Sorting. Searching strategies. Weighted graphs. The shortest path and travelling salesman problems. Networks and flows. References and further reading. Hints and solutions to selected exercises. Index.
"There is now a wide agreement that mathematics for computer scientists is not calculus and numerical analysis but discrete mathematics. According to this idea this book is a lovely text for undergraduate students, useful also at freshman-sophomore level."
-Zentralblatt für Mathematik und ihre Grenzgebiete
"Discrete Mathematics for New Technology is a nice introduction to logic, sets, and algebraic structures."
"The book has a number of noteworthy features which make it a text to be considered seriously by those who wish to teach a course in this area. The material is carefully written and clearly presented in a user-friendly way which makes it a pleasure to read. There is a wealth of well-judged examples, with frequent historical notes to provide background and cartoons to lighten the style."
-Times Higher Education Supplement