Discrete Mathematics

1. Front Matter

   Pages 1-1

   PDF

2. Mathematical Reasoning, Proof Principles, and Logic

   • Jean Gallier

   Pages 1-100

3. Relations, Functions, Partial Functions

   • Jean Gallier

   Pages 101-164

4. Graphs, Part I: Basic Notions

   • Jean Gallier

   Pages 165-203

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

   • Jean Gallier

   Pages 205-255

6. Partial Orders, Lattices, Well-Founded 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

   • Jean Gallier

   Pages 257-363

7. Graphs, Part II: More Advanced Notions

   • Jean Gallier

   Pages 365-447

8. Back Matter

   Pages 446-446

   PDF

This books gives an introduction to discrete mathematics for beginning undergraduates. One of original features of this book is that it begins with a presentation of the rules of logic as used in mathematics. 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 is more standard. It deals with functions and relations, directed and undirected graphs, and an introduction to combinatorics. 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. Eulerian and Hamiltonian cycles are discussed. Then, we study flows and tensions and state and prove the max flow min-cut theorem. We also discuss matchings, covering, bipartite graphs.

• combinatorics
• graphs
• logic
• proofs
• public key cryptography

From the reviews:

“This well-written, 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, end-of-chapter summaries and references, a useful set of problems of varying difficulty, and a symbol as well as a subject index. Summing Up: Highly recommended. Upper-division 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)

• , Computer and Information Science, University of Pennsylvania, Philadelphia, USA

  Jean Gallier

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