 Summarizing the rules of mathematical reasoning and how to construct proofs
 Presents examples of formal and informal proofs
 Includes examples of proofs by induction
 Discusses public key cryptography, with a complete proof of the correctness of RSA Explicit, detailed algorithms for modular arithmetic
 Explores graph flows and the maxflow mincut theorem
 Covers planar graphs
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 About this Textbook

This book gives an introduction to discrete mathematics for beginning undergraduates and starts with a chapter on the rules of mathematical reasoning.
This book begins with a presentation of the rules of logic as used in mathematics where many examples of formal and informal proofs are given. With this logical framework firmly in place, the book describes the major axioms of set theory and introduces the natural numbers. The rest of the book deals with functions and relations, directed and undirected graphs and an introduction to combinatorics, partial orders and complete induction. There is a section on public key cryptography and RSA, with complete proofs of Fermat's little theorem and the correctness of the RSA scheme, as well as explicit algorithms to perform modular arithmetic. The last chapter provides more graph theory where Eulerian and Hamiltonian cycles are discussed. This book also includes network flows, matchings, covering, bipartite graphs, planar graphs and state the graph minor theorem of Seymour and Robertson.
The book is highly illustrated and each chapter ends with a list of problems of varying difficulty. Undergraduates in mathematics and computer science will find this book useful.
 About the authors

Jean Gallier is a Professor in the Computer and Information Science Department, School of Engineering and Applied Science at the University of Pennsylvania.
 Reviews

From the reviews:
“This wellwritten, highly illustrated book will be very useful and interesting to students in both mathematics and computer science. … Attractive features of this book include clear presentations, endofchapter summaries and references, a useful set of problems of varying difficulty, and a symbol as well as a subject index. Summing Up: Highly recommended. Upperdivision undergraduates, graduate students, and professionals/practitioners.” (D. V. Chopra, Choice, Vol. 48 (11), July, 2011)
“This book is intended to be a textbook for students in Computer Science, covering basic areas of Discrete Mathematics. … lots of references to supplementary or more advanced literature are provided, and less basic and more sophisticated problems as well as connections to other areas of science are given. Each chapter closes with a rich collection of exercises, which often include hints to their solution and further explanations.” (Martina Kubitzke, Zentralblatt MATH, Vol. 1227, 2012)
“This book provides a rigorous introduction to standard topics in the field: logical reasoning, sets, functions, graphs and counting techniques. Its intended audience is computer science undergraduate students, but could also be used in a course for mathematics majors. … Each chapter has a summary and a generous number of exercises … . The exposition is structured as a series of propositions and theorems that are proved clearly and in detail. Historical remarks and an abundance of photographs of mathematicians enliven the text.” (Gabriella Pinter, The Mathematical Association of America, February, 2012)
 Table of contents (6 chapters)


Mathematical Reasoning, Proof Principles, and Logic
Pages 1100

Relations, Functions, Partial Functions
Pages 101164

Graphs, Part I: Basic Notions
Pages 165203

Some Counting Problems; Multinomial Coefficients, The Principle of Inclusion–Exclusion, Sylvester’s Formula, The Sieve Formula
Pages 205255

Partial Orders, Lattices, WellFounded Orderings, Unique Prime Factorization in ℤ and GCDs, Equivalence Relations, Fibonacci and Lucas Numbers, Public Key Cryptography and RSA, Distributive Lattices, Boolean Algebras, Heyting Algebras
Pages 257363

Table of contents (6 chapters)
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Bibliographic Information
 Bibliographic Information

 Book Title
 Discrete Mathematics
 Authors

 Jean Gallier
 Series Title
 Universitext
 Copyright
 2011
 Publisher
 SpringerVerlag New York
 Copyright Holder
 Springer Science+Business Media, LLC
 eBook ISBN
 9781441980472
 DOI
 10.1007/9781441980472
 Softcover ISBN
 9781441980465
 Series ISSN
 01725939
 Edition Number
 1
 Number of Pages
 XIV, 466
 Number of Illustrations
 200 b/w illustrations, 20 illustrations in colour
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