Discrete Mathematics With Proof

This book has been written for a sophomore-level course in Discrete Mathematics. The material has been directed towards the needs of mathematics and computer science majors, although there is certainly material that is of use for other majors. Students are assumed to have completed a semester of college-level calculus. This assumption is primarily about the level of mathematical maturity of the readers. The material in a calculus course will not often be used in the text. This textbook has been designed to be suitable for a course that requires students to read the textbook. Many students find this challenging, preferring to just let the instructor tell them "everything they need to know" and using the textbook as a repository of homework exercises and corresponding examples. A typical course in Discrete Mathematics will require much more from the students. Consequently, the textbook needs to support this transition towards greater mathematical maturity. I have successfully used this text by requiring students to read a section and submit some simple exercises from that section at the start of a class period where I discuss the material for the first time. The following class period, the students will submit more difficult exercises. Consequently, extra care has been taken to ensure that students can follow the presentation in the book even before the material is presented in class. While most instructors do not structure their course in this manner, a textbook that has been written to stand on its own will certainly be of value to the students. I imagine that this book will work well with a distance education format. However, I feel that personal interaction between the student and the instructor (or a knowledgeable teaching assistant) greatly enhances the learning experience. DISTINGUISHING CHARACTERISTICS OF THIS TEXT There are currently many textbooks on the market for a course in Discrete Mathematics. Although there is an assumed common core of topics and level, there is still sufficient variation to provide instructors with viable options for choosing a textbook. Here are some of the features that characterize this book. There is a heavy emphasis on proof throughout the text (as indicated by the book's title). The formal setting is introduced in Chapter 2 as sets, logic, and Boolean algebras are discussed. Chapter 3 then discusses axiomatic mathematics as a system and subsequently focuses on proof techniques. The proof techniques are extensively illustrated in the rest of the text. For example: proof by contradiction in Chapter 4 with The Halting Problem; constructive proofs in Chapter 8 with "a finite projective plane of orderniffn-- 1 mutually orthogonal Latin squares of ordern"; complete induction in Chapter 3 with the "optimality (for suitors) of the Deferred Acceptance Algorithm". Combinatorial proof is introduced in Chapter 5 and used in Chapter 8 to establish the necessary conditions for the existence of a balanced incomplete block design. Many of the more difficult proofs are accompanied by illustrative examples that can be read in parallel with the proof. For instance, Theorem 8.20 on page 486 and Examples 8.49 and 8.50 that appear after the proof. The text has been written for students to read actively. The text contains more detailed explanations than some competing texts. Homework problems have been designed to reinforce reading. Most cannot be completed by merely finding a clone example to copy and modify. The chapters include Quick Check problems at critical points in the reading. These are problems that should be solved before continuing to read. Detailed solutions are presented at the end of the chapter. Technology is introduced when it will enhance understanding. For example: a simple perl script for testing regular expressions in Chapter 9; a Java Application (and Applet) that allows studentGossett, Eric is the author of 'Discrete Mathematics With Proof', published 2002 under ISBN 9780130669483 and ISBN 0130669482.