Discrete Mathematics
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Discrete Mathematics
This gentle introduction to discrete mathematics is written for first and second year math majors, especially those who intend to teach. The text began as a set of lecture notes for the discrete mathematics course at the University of Northern Colorado. This course serves both as an introduction to topics in discrete math and as the "introduction to proof" course for math majors. The course is usually taught with a large amount of student inquiry, and this text is written to help facilitate this. Four main topics are covered: counting, sequences, logic, and graph theory. Along the way proofs are introduced, including proofs by contradiction, proofs by induction, and combinatorial proofs. The book contains over 360 exercises, including 230 with solutions and 130 more involved problems suitable for homework. There are also Investigate! activities throughout the text to support active, inquiry based learning. While there are many fine discrete math textbooks available, this text has the following advantages: It is written to be used in an inquiry rich course. It is written to be used in a course for future math teachers. It is open source, with low cost print editions and free electronic editions. Update: as of July 2017, this 2nd edition has been updated, correcting numerous typos and a few mathematical errors. Pagination is almost identical to the earlier printing of the 2nd edition. For a list of changes, see the book's website: http: //discretetext.oscarlevin.com
Discrete Mathematics for Computer Science
Discrete Mathematics for Computer Science by Gary Haggard , John Schlipf , Sue Whitesides A major aim of this book is to help you develop mathematical maturity-elusive as thisobjective may be. We interpret this as preparing you to understand how to do proofs ofresults about discrete structures that represent concepts you deal with in computer science.A correct proof can be viewed as a set of reasoned steps that persuade another student,the course grader, or the instructor about the truth of the assertion. Writing proofs is hardwork even for the most experienced person, but it is a skill that needs to be developedthrough practice. We can only encourage you to be patient with the process. Keep tryingout your proofs on other students, graders, and instructors to gain the confidence that willhelp you in using proofs as a natural part of your ability to solve problems and understandnew material. The six chapters referred to contain the fundamental topics. Thesechapters are used to guide students in learning how to express mathematically precise ideasin the language of mathematics.The two chapters dealing with graph theory and combinatorics are also core materialfor a discrete structures course, but this material always seems more intuitive to studentsthan the formalism of the first four chapters. Topics from the first four chapters are freelyused in these later chapters. The chapter on discrete probability builds on the chapter oncombinatorics. The chapter on the analysis of algorithms uses notions from the core chap-ters but can be presented at an informal level to motivate the topic without spending a lot oftime with the details of the chapter. Finally, the chapter on recurrence relations primarilyuses the early material on induction and an intuitive understanding of the chapter on theanalysis of algorithms. The material in Chapters 1 through 4 deals with sets, logic, relations, and functions.This material should be mastered by all students. A course can cover this material at differ-ent levels and paces depending on the program and the background of the students whenthey take the course. Chapter 6 introduces graph theory, with an emphasis on examplesthat are encountered in computer science. Undirected graphs, trees, and directed graphsare studied. Chapter 7 deals with counting and combinatorics, with topics ranging from theaddition and multiplication principles to permutations and combinations of distinguishableor indistinguishable sets of elements to combinatorial identities.Enrichment topics such as relational databases, languages and regular sets, uncom-putability, finite probability, and recurrence relations all provide insights regarding howdiscrete structures describe the important notions studied and used in computer science.Obviously, these additional topics cannot be dealt with along with the all the core materialin a one-semester course, but the topics provide attractive alternatives for a variety of pro-grams. This text can also be used as a reference in courses. The many problems provideample opportunity for students to deal with the material presented.
Discrete Mathematics
Author: Jean Gallier
language: en
Publisher: Springer Science & Business Media
Release Date: 2011-02-01
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.