Mathematical logic, set theory, the real numbers, induction and recursion, summation notation, asymptotic notation, number theory, relations, graphs, counting, linear algebra, finite fields. Free discrete mathematics books download ebooks online. A logical argument establishing the truth of the theorem given the truth of the axioms and any previously proven theorems. Discrete mathematics lecture notes, yale university, spring 1999 l. Discrete mathematics uses a range of techniques, some of which is seldom found in its continuous counterpart. This book presents not only the major themes of discrete mathematics, but also the reasoning that underlies mathematical thought. Many different systems of axioms have been proposed. The author uses a range of examples to teach core concepts, while corresponding exercises allow students to apply what they learn.
Mathematical proofs can themselves be represented formally as discrete structures. Discrete mathematics discrete mathematics study of mathematical structures and objects that are fundamentally discrete rather than continuous. Richard mayr university of edinburgh, uk discrete mathematics. Discrete math basic proof methods overview of basic proof methods 1. A binary relation from a to b is a subset of a cartesian product a x b. Direct proof example 2 show that if m and n are both square numbers, then m n is also a square number.
In many proofs, one needs to combine several different strategies. Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. A course emphasizing discrete mathematics could cover chapters 112. Discrete mathematics combines a balance of theory and applications with mathematical rigor and an accessible writing style. Johnsonbaugh, discrete mathematics, 8th edition pearson. R tle a x b means r is a set of ordered pairs of the form a,b where a a and b b. Discrete mathematics is the branch of mathematics in which we. Mathematics for computer science eric lehman and tom leighton 2004. Discrete individually separate and distinct as opposed to continuous and capable of infinitesimal change.
Renowned for her lucid, accessible prose, epp explains complex, abstract concepts with clarity and precision. We are going to apply the logical rules in proving mathematical theorems. This is a course on discrete mathematics as used in computer science. Proofs are to mathematics what spelling is to poetry. Combinatorics, graph theory, the idea of function, recurrence relations, permutations, and set theory are all part of discrete math. A course in discrete structures cornell university. This implies that there are integers u and v such that m u2 and n v2. 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. The development of some material on this site was funded by nsf due0230755. Given two statements p and q, there are several ways to combine them to get a new. This is a book about discrete mathematics which also discusses mathematical rea soning and.
It is increasingly being applied in the practical fields of mathematics and computer science. It is important to realize that there is no mathematics without proofs. Johnsonbaugh, discrete mathematics, global edition pearson. Number theory athe math behind the rsa crypto system. In this video we tackle a divisbility proof and then prove that all integers are the difference of two squares. An introduction to mathematical reasoning, or those using any other textbook. Discrete mathematics consists of many individual topics that, imprecisely, contrast with continuous mathematics, e. Think of digital watches versus analog watches ones where the. In contrast, continuous mathematics deals with objects that vary continuously, e.
But a sharp contrast between discrete and continuous mathematics is. This lecture now we have learnt the basics in logic. Discrete mathematics direct proofs examples youtube. 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 structures lecture notes stanford university. Discrete structures include sets, permutations, graphs, trees, variables in computer programs, and finitestate machines. Exercise sets features a large number of applications. Sequences and series are among the most important applications of. 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. Richard johnsonbaugh is professor emeritus of computer science, telecommunications and information systems, depaul university, chicago. Direct proof contrapositive proof by contradiction proof by cases 3. This course will roughly cover the following topics and speci c applications in computer science. 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. Proofs, not computations, form the core of real mathematics.
Download discrete mathematics pdf ebook discrete mathematics discrete mathematics ebook author by sarahmarie belcastro. The book begins with an introductory chapter that provides an accessible explanation of. Discrete mathematics proof by cases or proof by elimination of cases. Csc 201 discrete mathematics for computer scientists. The text covers the mathematical concepts that students will encounter in many disciplines such as computer science, engineering, business, and the sciences.
Proofs by mathematical induction do not always start at the integer 0. Browse other questions tagged discretemathematics proofexplanation or ask your own question. The twosemester discrete math sequence covers the mathematical topics most directly related to computer science. Mathematical reasoning and proof with puzzles, patterns and games, by doug ensley and winston crawley, published by john wiley and sons. Students develop the ability to think abstractly as they study the ideas of logic and proof. Discrete mathematics tutorial in pdf tutorialspoint. The material is o ered as the secondyear course comp 2804 discrete structures ii. Merely stating the facts, without saying something about why these facts are valid, would be terribly far from the spirit. Discrete mathematics amit chakrabarti proofs by contradiction and by mathematical induction direct proofs at this point, we have seen a few examples of mathematicalproofs. Richard johnsonbaugh, depaul university 2019 pearson format paper isbn. In such a case, the basis step begins at a starting point b where b is an integer. This is a course note on discrete mathematics as used in computer science. The discrete mathematics course at bethel college is equally populated with mathematics majors and computer science majors.
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. Besides reading the book, students are strongly encouraged to do all the. In this case we prove the property only for integers b instead of for all of n. Consequently, this text was designed to be appropriate for courses for mathematics majors, courses for computer science majors, and courses with bimodal populations. Problems on discrete mathematics1 ltex at january 11, 2007. Discrete mathematics, second edition in progress upenn cis. Discrete mathematics i about the tutorial discrete mathematics is a branch of mathematics involving discrete elements that uses algebra and arithmetic. Issues about data structures used to represent sets and the computational cost of set operations.
A proof is a valid argument that establishes the truth of a theorem as the conclusion. Methods of proving common mistakes in proofs strategies. 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. Mathematics for computer science eric lehman and tom. Discrete mathematics deals with objects that come in discrete bundles, e. This book is designed for a one semester course in discrete mathematics for sophomore or junior level students.
Vesztergombi parts of these lecture notes are based on l. Examples of objectswith discrete values are integers, graphs, or statements in logic. Combining as it does the six most fundamental constants of mathematics. A direct proof is a mathematical argument that uses rules of inference to derive the conclusion from the premises. Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields. 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. Discrete mathematics mathematical proof techniques.
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