Although the sixth edition has been an extremely effective text, many instructors, including longtime users, have requested changes designed to make this book more effective. I have devoted a significant amount of time and energy to satisfy their requests and I have worked hard to find my own ways to make the book more effective and more compelling to students.

The seventh edition is a major revision, with changes based on input from more than 40 formal reviewers, feedback from students and instructors, and author insights. The result is a new edition that offers an improved organization of topics making the book a more effective teaching tool. Substantial enhancements to the material devoted to logic, algorithms, number theory, and graph theory make this book more flexible and comprehensive. Numerous changes in the seventh edition have been designed to help students more easily learn the material. Additional explanations and examples have been added to clarify material where students often have difficulty. New exercises, both routine and challenging, have been added. Highly relevant applications, including many related to the Internet, to computer science, and to mathematical biology, have been added. The companion website has benefited from extensive development activity and now provides tools students can use to master key concepts and explore the world of discrete mathematics, and many new tools under development will be released in the year following publication of this book.

I hope that instructors will closely examine this new edition to discover how it might meet their needs. Although it is impractical to list all the changes in this edition, a brief list that highlights some key changes, listed by the benefits they provide, may be useful.

More Flexible Organization

• Applications of propositional logic are found in a new dedicated section, which briefly introduces logic circuits.
• Recurrence relations are now covered in Chapter 2.
• Expanded coverage of countability is now found in a dedicated section in Chapter 2.
• Separate chapters now provide expanded coverage of algorithms (Chapter 3) and number theory and cryptography (Chapter 4).
• More second and third level heads have been used to break sections into smaller coherent parts.

Tools for Easier Learning

• Difficult discussions and proofs have been marked with the famous Bourbaki “dangerous bend” symbol in the margin.
• New marginal notes make connections, add interesting notes, and provide advice to students.
• More details and added explanations, in both proofs and exposition, make it easier for students to read the book.
• Many new exercises, both routine and challenging, have been added, while many existing exercises have been improved.

Enhanced Coverage of Logic, Sets, and Proof

• The satisfiability problem is addressed in greater depth, with Sudoku modeled in terms of satisfiability.
• Hilbert’s Grand Hotel is used to help explain uncountability.
• Proofs throughout the book have been made more accessible by adding steps and reasons behind these steps.
• A template for proofs by mathematical induction has been added.
• The step that applies the inductive hypothesis in mathematical induction proof is now explicitly noted.

Algorithms

• The pseudocode used in the book has been updated.
• Explicit coverage of algorithmic paradigms, including brute force, greedy algorithms, and dynamic programing, is now provided.
• Useful rules for big-O estimates of logarithms, powers, and exponential functions have been added.

Number Theory and Cryptography

• Expanded coverage allows instructors to include just a little or a lot of number theory in their courses.
• The relationship between the mod function and congruences has been explained more fully.
• The sieve of Eratosthenes is now introduced earlier in the book.
• Linear congruences and modular inverses are now covered in more detail.
• Applications of number theory, including check digits and hash functions, are covered in great depth.
• A new section on cryptography integrates previous coverage, and the notion of a cryptosystem has been introduced.
• Cryptographic protocols, including digital signatures and key sharing, are now covered.

Graph Theory

• A structured introduction to graph theory applications has been added.
• More coverage has been devoted to the notion of social networks.
• Applications to the biological sciences and motivating applications for graph isomorphism and planarity have been added.
• Matchings in bipartite graphs are now covered, including Hall’s theorem and its proof.
• Coverage of vertex connectivity, edge connectivity, and n-connectedness has been added, providing more insight into the connectedness of graphs.

Enrichment Material

• Many biographies have been expanded and updated, and new biographies of Bellman, Bézout Bienyamé, Cardano, Catalan, Cocks, Cook, Dirac, Hall, Hilbert, Ore, and Tao have been added.
• Historical information has been added throughout the text.
• Numerous updates for latest discoveries have been made.

Expanded Media

• Extensive effort has been devoted to producing valuable web resources for this book.
• Extra examples in key parts of the text have been provided on companion website.
• Interactive algorithms have been developed, with tools for using them to explore topics and for classroom use.
• A new online ancillary, The Virtual Discrete Mathematics Tutor, available in fall 2012, will help students overcome problems learning discrete mathematics.
• A new homework delivery system, available in fall 2012, will provide automated homework for both numerical and conceptual exercises.
• Student assessment modules are available for key concepts.
• Powerpoint transparencies for instructor use have been developed.
• A supplement Exploring Discrete Mathematics has been developed, providing extensive support for using Maple^TM or Mathematica^TM in conjunction with the book.
• An extensive collection of external web links is provided.