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Discrete Mathematics
Proofs, Structures and Applications, Third Edition
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Book Description
Taking an approach to the subject that is suitable for a broad readership, Discrete Mathematics: Proofs, Structures, and Applications, Third Edition provides a rigorous yet accessible exposition of discrete mathematics, including the core mathematical foundation of computer science. The approach is comprehensive yet maintains an easytofollow progression from the basic mathematical ideas to the more sophisticated concepts examined later in the book. This edition preserves the philosophy of its predecessors while updating and revising some of the content.
New to the Third Edition
In the expanded first chapter, the text includes a new section on the formal proof of the validity of arguments in propositional logic before moving on to predicate logic. This edition also contains a new chapter on elementary number theory and congruences. This chapter explores groups that arise in modular arithmetic and RSA encryption, a widely used public key encryption scheme that enables practical and secure means of encrypting data. This third edition also offers a detailed solutions manual for qualifying instructors.
Exploring the relationship between mathematics and computer science, this text continues to provide a secure grounding in the theory of discrete mathematics and to augment the theoretical foundation with salient applications. It is designed to help readers develop the rigorous logical thinking required to adapt to the demands of the everevolving discipline of computer science.
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
Arguments
Formal Proof of the Validity of 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
Types and Typed Set Theory
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
Introduction to Number Theory
Divisibility
Prime Numbers
Linear Congruences
Groups in Modular Arithmetic
Public Key Cryptography
Boolean Algebra
Introduction
Properties of Boolean Algebras
Boolean Functions
Switching Circuits
Logic Networks
Minimization of Boolean Expressions
Graph Theory
Definitions and Examples
Paths and Cycles
Isomorphism of Graphs
Trees
Planar Graphs
Directed Graphs
Applications of Graph Theory
Introduction
Rooted Trees
Sorting
Searching Strategies
Weighted Graphs
The Shortest Path and Traveling Salesman Problems
Networks and Flows
References and Further Reading
Hints and Solutions to Selected Exercises
Index
Author(s)
Biography
Rowan Garnier was a professor of mathematics at Richmond, the American International University in London, where she served ten years as Chair of the Division of Mathematics, Science and Computer Science.
John Taylor is Head of the School of Computing, Mathematical and Information Sciences at the University of Brighton, UK. He has published widely on the applications of diagrammatic logic systems to computer science.
Reviews
The authors’ diligent attempt to present, analyse and thoroughly demonstrate the subject of DMths is noteworthy. In keeping with the textbook character of their book, they also cite many examples. The book is an integrated textbook of DMths, adequate for undergraduate computer scientists, featuring a synoptic and vital presentation of this important, useful and interesting field. Of course, it is also interesting and useful for students of mathematics, as well as for those who work with informatics in general. It is a classic textbook, well structured and sufficiently complete within the framework established by similar textbooks. The work does a good job keeping a balance between conciseness and indepth examinations … .
—Contemporary Physics, Vol. 52, No. 2, MarchApril 2011This is a textbook on discrete mathematics for undergraduate students in computer science and mathematics. The choice of the topics covered in this text is largely suggested by the needs of computer science. It contains chapters on set theory, logic, algebra (matrix algebra and Boolean algebra), and graph theory with applications. … The style of exposition is very clear, step by step and the level is well adapted to undergraduates in computer science. The treatment is mathematically rigorous; therefore it is also suitable for mathematics students. Besides the theory there are many concrete examples and exercises (with solutions!) to develop the routine of the student. So I can recommend warmly this book as a textbook for a course. It looks very attractive and has a nice typography. … Although I haven’t used this book in class (up to now), I think it is an excellent textbook.
—H.G.J. Pijls, University of Amsterdam, The NetherlandsPraise for Previous Editions
Garnier and Taylor offer a work on discrete mathematics sufficiently comprehensive to be used as a resource work in a variety of courses … Now in its second edition, it would also make an excellent general reference book on these areas … a fine undergraduate book.
—R.L. Pour, Emory & Henry College, CHOICE
Provides an accessible introduction to discrete mathematics, including the core mathematics requirements for undergraduate computer science students.
—SciTech Book News, Vol. 122
This is the second edition of this accessible yet rigorous introduction to discrete mathematics. As in the first edition, the theory is illustrated by a large number of solved exercises. In this edition further exercises have been added, in particular, at the routine level. In addition, some new material on typed set theory is included.
—S. Teschl
The book is designed for students of computer science. It contains main mathematical topics needed in their undergraduate study. In the second edition, the authors added a lot of new exercises and examples, illustrating discussed concepts. The book contains a lot of wellordered and nicely illustrated material.
—Vladimir Soucek, European Mathematical Society Newsletter, June 2004
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