This book equips readers to apply discrete mathematics and provides opportunities for practice of the concepts presented. Coverage of algorithms is included. Combinatorics receives more coverage than in other books.
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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.
The chapters in the book are briefly summarized in the following paragraphs.
Chapter 1: Introduction
Chapter h provides a working definition of discrete mathematics and then offers the reader some brief glimpses at some of the topics that will be covered in the remaining chapters. The chapter also introduces the stable marriage problem and the deferred acceptance algorithm. This material is covered in some detail and appears again in several other chapters.
The exposition of the stable marriage problem introduces a non-trivial algorithm and some proofs. The problem, the algorithm, and the proofs are all fairly intuitive. They prepare the reader for the more detailed expositions of algorithms and proofs that will follow in future chapters. The problem also shows the reader that the material in this course may be different from what they have studied in previous mathematics courses.
Chapter 2: Sets, Logic, and Boolean Algebras
Much of the material in this chapter is not what students tend to rate as most interesting. However, it is foundational to much of what follows. It is even more important than in previous decades because many students are now graduating from high school without ever learning the basics of set theory. Many have never been exposed to either the basic terminology (element, union, intersection) or the standard notation (E, U, f1).
The basic concepts of propositional and predicate logic are introduced in this chapter. They also serve as a basis for the proof strategies introduced in chapter 3.
The basic properties of sets and logic are presented in a similar style to emphasize the similarities. This parallel exposition provides a natural introduction to Boolean algebras. Boolean algebras serve to unify some important aspects of set theory and logic. The early introduction also provides a nontrivial example of an axiomatic system. This example can then be recalled when the axiomatic system is more formally introduced in chapter 3.
The chapter also contains brief sections on informal logic and analyzing claims. Both sections...
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