We are going to apply the logical rules in proving mathematical theorems. In such a case, the basis step begins at a starting point b where b is an integer. This book is designed for a one semester course in discrete mathematics for sophomore or junior level students. Mathematical reasoning and proof with puzzles, patterns and games, by doug ensley and winston crawley, published by john wiley and sons. Discrete mathematics is the branch of mathematics in which we. Discrete mathematics mathematical proof techniques. A binary relation from a to b is a subset of a cartesian product a x b.
A course in discrete structures cornell university. An introduction to mathematical reasoning, or those using any other textbook. In this case we prove the property only for integers b instead of for all of n. It is increasingly being applied in the practical fields of mathematics and computer science. This implies that there are integers u and v such that m u2 and n v2.
This is a course on discrete mathematics as used in computer science. Johnsonbaugh, discrete mathematics, global edition pearson. Discrete mathematics, second edition in progress upenn cis. Many different systems of axioms have been proposed.
A course emphasizing discrete mathematics could cover chapters 112. The author uses a range of examples to teach core concepts, while corresponding exercises allow students to apply what they learn. Merely stating the facts, without saying something about why these facts are valid, would be terribly far from the spirit. In many proofs, one needs to combine several different strategies. Renowned for her lucid, accessible prose, epp explains complex, abstract concepts with clarity and precision. This webpage was created as a resource for students of discrete mathematics, either those using one of my textbooks, discrete mathematics with applications, 4 th edition, or discrete mathematics. Combinatorics, graph theory, the idea of function, recurrence relations, permutations, and set theory are all part of discrete math. Richard johnsonbaugh, depaul university 2019 pearson format paper isbn. Direct proof contrapositive proof by contradiction proof by cases 3. Richard johnsonbaugh is professor emeritus of computer science, telecommunications and information systems, depaul university, chicago.
R tle a x b means r is a set of ordered pairs of the form a,b where a a and b b. Proofs by mathematical induction do not always start at the integer 0. Besides reading the book, students are strongly encouraged to do all the. Mathematical logic, set theory, the real numbers, induction and recursion, summation notation, asymptotic notation, number theory, relations, graphs, counting, linear algebra, finite fields. Mathematical proofs can themselves be represented formally as discrete structures. With nearly 4,500 exercises, discrete mathematics provides ample opportunities for students to practice, apply, and demonstrate conceptual understanding. In this video we tackle a divisbility proof and then prove that all integers are the difference of two squares. Vesztergombi parts of these lecture notes are based on l. An indirect proof uses rules of inference on the negation of the conclusion and on some of the premises to derive the negation of a premise. Discrete mathematics i about the tutorial discrete mathematics is a branch of mathematics involving discrete elements that uses algebra and arithmetic. Discrete mathematics with proof, second edition continues to facilitate an uptodate understanding of this important topic, exposing readers to a wide range of modern and technological applications. Discrete math concerns counting, probability, sophisticated forms of addition, and limit processes over discrete sets.
Exercise sets features a large number of applications. It is important to realize that there is no mathematics without proofs. The discrete mathematics course at bethel college is equally populated with mathematics majors and computer science majors. But a sharp contrast between discrete and continuous mathematics is. This is a book about discrete mathematics which also discusses mathematical rea soning and. An accessible introduction to the topics of discrete math, this bestselling text also works to expand students mathematical maturity.
Discrete mathematics deals with objects that come in discrete bundles, e. A direct proof is a mathematical argument that uses rules of inference to derive the conclusion from the premises. Discrete individually separate and distinct as opposed to continuous and capable of infinitesimal change. Given two statements p and q, there are several ways to combine them to get a new. Discrete mathematics direct proofs examples youtube. Combining as it does the six most fundamental constants of mathematics. Csc 201 discrete mathematics for computer scientists. These applets accompany the textbook, discrete mathematics. Prior to his 20year service at depaul university, he was a member and sometime chair of the mathematics departments at morehouse college and chicago state university. The book begins with an introductory chapter that provides an accessible explanation of. Free discrete mathematics books download ebooks online. Direct proof example 2 show that if m and n are both square numbers, then m n is also a square number.
Issues about data structures used to represent sets and the computational cost of set operations. This course will roughly cover the following topics and speci c applications in computer science. The development of some material on this site was funded by nsf due0230755. Browse other questions tagged discretemathematics proofexplanation or ask your own question. This lecture now we have learnt the basics in logic. Methods of proving common mistakes in proofs strategies. Consequently, this text was designed to be appropriate for courses for mathematics majors, courses for computer science majors, and courses with bimodal populations. Discrete mathematics tutorial in pdf tutorialspoint. This is a course note on discrete mathematics as used in computer science. Mathematics for computer science eric lehman and tom leighton 2004.
Sequences and series are among the most important applications of. Students develop the ability to think abstractly as they study the ideas of logic and proof. Problems on discrete mathematics1 chungchih li2 kishan mehrotra3 syracuse university, new york latex at january 11, 2007 part i 1no part of this book can be reproduced without permission from the authors. Discrete mathematics proof by cases or proof by elimination of cases. Discrete mathematics discrete mathematics study of mathematical structures and objects that are fundamentally discrete rather than continuous.
The twosemester discrete math sequence covers the mathematical topics most directly related to computer science. Johnsonbaugh, discrete mathematics, 8th edition pearson. Discrete mathematics amit chakrabarti proofs by contradiction and by mathematical induction direct proofs at this point, we have seen a few examples of mathematicalproofs. In contrast to real numbers that have the property of varying smoothly, the objects studied in discrete mathematics such as integers, graphs, and statements in logic do not vary smoothly in this way, but have distinct, separated values. A logical argument establishing the truth of the theorem given the truth of the axioms and any previously proven theorems. Think of digital watches versus analog watches ones where the. Basic building block for types of objects in discrete mathematics. A proof is a valid argument that establishes the truth of a theorem as the conclusion. Discrete mathematics uses a range of techniques, some of which is seldom found in its continuous counterpart. Richard mayr university of edinburgh, uk discrete mathematics. Discrete math basic proof methods overview of basic proof methods 1.
Examples of objectswith discrete values are integers, graphs, or statements in logic. The text covers the mathematical concepts that students will encounter in many disciplines such as computer science, engineering, business, and the sciences. In contrast, continuous mathematics deals with objects that vary continuously, e. Discrete structures include sets, permutations, graphs, trees, variables in computer programs, and finitestate machines. Discrete mathematics consists of many individual topics that, imprecisely, contrast with continuous mathematics, e. The material is o ered as the secondyear course comp 2804 discrete structures ii. Discrete structures lecture notes stanford university. Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. Mathematics for computer science eric lehman and tom. This book presents not only the major themes of discrete mathematics, but also the reasoning that underlies mathematical thought. Proofs are to mathematics what spelling is to poetry.
This is a free textbook for an undergraduate course on discrete structures for computer science students, which i have been teaching at carleton university since the fall term of 20. Problems on discrete mathematics1 ltex at january 11, 2007. Discrete mathematics richard johnsonbaugh download. Discrete mathematics lecture notes, yale university, spring 1999 l. Number theory athe math behind the rsa crypto system. Download discrete mathematics pdf ebook discrete mathematics discrete mathematics ebook author by sarahmarie belcastro.
1146 366 985 1005 598 330 316 1038 481 929 1011 1083 1558 223 1002 219 1262 1082 1008 613 660 615 1513 1056 1449 1045 631 1041 1490 1438 749 907