Discrete structures lecture notes vladlen koltun1 winter 2008 1computer science department, 353 serra mall, gates 374, stanford university, stanford, ca 94305, usa. However, the usual professor is on leave and has been replaced by one who absolutely cannot teach. A set is a welldefined collection of distinct objects. Paces mathematics of discrete structures for computer science is a delightful book, written to be read and enjoyed as it moves from justifying the central place of maths in computing, through the usual set theory and logic, on to numbers, types and the halting problem. Applied discrete structures has been developed using pretext, a lightweight. The goal of this course is to provide a background in these topics. The text covers the mathematical concepts that students will encounter in many disciplines such as computer science, engineering, business, and the sciences. Besides reading the book, students are strongly encouraged to do all the. The order of the elements in a set doesnt contribute. R tle a x b means r is a set of ordered pairs of the form a,b where a a and b b. Real numbers have the property that between any two numbers a third can be found, and consequently these numbers vary smoothly.
This book has been written to fulfill the requirements of graduate and. A set is an unordered collection of distinct objects. A read is counted each time someone views a publication summary such as the title, abstract, and list of authors, clicks on a figure, or views or downloads the fulltext. Cs381 discrete structures discrete mathematics web course material last update august 2, 2009 note. This book is meant to be more than just a text in discrete mathematics. He had defined a set as a collection of definite and distinguishable objects selected by the mean. We only give an informal description of sets and then proceed to establish their properties. Basic concepts of set theory, functions and relations. This is a course note on discrete mathematics as used in computer science.
Today we introduce set theory, elements, and how to build sets. 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. This introductory section revisits ideas met in the early part of analysis i and in linear algebra i, to set the scene and provide. Discrete structures introduces readers to the mathematical structures and methods that form the foundation of computer science and features multiple techniques that readers will turn to regularly throughout their careers in computer and information sciences. Although elementary set theory is wellknown and straightforward, the modern subject, axiomatic set theory, is both conceptually more di. The ultimate goal of the two books are to make a strong case for the inclusion of discrete mathematics in the undergraduate curricula of mathematics by creating a sequence of courses in discrete mathematics parallel to the traditional.
In contrast to real numbers that vary smoothly, discrete mathematics studies objects such as integers, graphs, and statements in logic. A set can be defined as a unordered collection of entities that are related because they obey a certain rule. I promise that after reading this post, you will love discrete math mathematical induction this principle is simple. What are some of the best books on discrete math for. This video is an updated version of the original video released over two years ago. Discrete mathematics i about the tutorial discrete mathematics is a branch of mathematics involving discrete elements that uses algebra and arithmetic. Mathematical logic, set theory, the real numbers, induction and recursion, summation notation, asymptotic notation, number theory, relations, graphs, counting, linear algebra, finite fields. Our study of discrete structures has given us the background to develop these topics in instruction paragraphs. Discrete mathematics is in contrast to continuous mathematics, which deals with structures which can range in value over the real numbers, or.
This study guide is written to accompany discrete structures, logic, and computability, third edition, by james l. A binary relation from a to b is a subset of a cartesian product a x b. It is a forerunner of another book applied discrete structures by the same author. Discrete mathematics simple english wikipedia, the free. In this book, we will consider the intuitive or naive view point of sets. Spiral workbook for discrete mathematics open textbook. I am starting a 2nd year course in discrete math this semester. It is a very good tool for improving reasoning and problemsolving capabilities. Sets and set operations cs 441 discrete mathematics for cs m. Discrete structures provide a tool box in computer science. The objects in a set are called the elements, or members, of the set. In fact earlier editions of this book sided with those who.
Comprehensive, book style, notes not repackaged overheads. Spiral workbook for discrete mathematics open textbook library. Discrete mathematics uses a range of techniques, some of which is seldom found in its continuous counterpart. The aim of this book is not to cover discrete mathematics in depth it should be clear from the description above that such a task would be illde. Discrete here is used as the opposite of continuous. A book of set theory, first published by dover publications, inc. Discrete mathematicsset theory wikibooks, open books.
Basic sets, comprehension, indexed sets, unions, intersections, products, disjoint unions, powersets. This book is designed for a one semester course in discrete mathematics for sophomore or junior level students. These objects do not vary smoothly, but have distinct, separated values. Discrete mathematicsset theory wikibooks, open books for.
Discrete mathematicsfunctions and relations wikibooks. Number theory athe math behind the rsa crypto system. Issues about data structures used to represent sets and the computational cost of set operations. As opposed to the study of calculus or real numbers which deal with continuous variables, discrete structures deals with graphs and statements in logic which can be enumerated through the implementation of integers. We say that a is afactorordivisorof b and b is amultipleof a. If ais a set and ais an element of it, we write a2a. However, my experience when teaching cse260 was that 70% of the students enjoyed the logic material, as it reminded them of programming. This page intentionally left blank university of belgrade. We say that a is equaltob notationa bifandonlyifeveryelementofa isanelementof. Lecture notes in discrete mathematics download book. A \wellde ned collection of distinct objects can be considered to be a set. Ling 310, adapted from umass ling 409, partee lecture notes march 1, 2006 p. This definition and that of regular sets and regular expressions requires careful contemplation on your part.
An important feature of a set is that its elements are \distinct or \uniquely identi able. These are notes on discrete mathematics for computer scientists. Examples of structures that are discrete are combinations, graphs, and logical statements. A relation is any association or link between elements of one set, called the domain or less formally the set of inputs, and another set, called the range or set of outputs. Therecursive stepgives the rules for forming new elements in the set from those already known to be in the set. I this is why bijections are also calledinvertible functions instructor. A course in discrete structures cornell university. For example, the numbers 2, 4, and 6 are distinct objects when considered separately, but when they are considered collectively they form a single set of size three, written 2, 4, 6. Advance discrete structure is a compulsory paper in most of computing programs m. 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 for computer science some notes.
Aslam rajmohan rajaraman eric ropiak chris burrows ravi sundaram college of computer and information science northeastern university boston, massachusetts. It is also important to be familiar with and be able to prove statements about sets. Many different systems of axioms have been proposed. Acknowledgements this book would not exist if not for discrete and combinatorial mathematics by richard grassl and tabitha mingus. Free discrete mathematics books download ebooks online.
Problems on discrete mathematics1 ltex at january 11, 2007. Reasonable efforts have been made to test interactive exercises and tools that have been developed here. Discrete mathematics therefore excludes topics in continuous mathematics such as. The book is intended for sophomore or junior level, corresponding to the course b3, introduction to discrete structures, in the acm curriculum 68, as well as for mathematicians or professors of computer engineering and advanced mathematics. Complex issues arise in set theory more than any other area of pure mathematics. I think its much more suitable as a second discrete math book. It explains and clarifies the unwritten conventions in mathematics, and guides the students through a detailed discussion on how a proof is revised from its. Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. In mathematics, a set is a welldefined collection of distinct objects, considered as an object in its own right. For example, the set 2,4,17,23 is the same as the set 17,4,23,2. Hauskrecht basic discrete structures discrete math study of the discrete structures used to represent discrete objects many discrete structures are built using sets sets collection of objects examples of discrete structures built with the help of sets.
Cantors diagonal argument to show powerset strictly increases size. Available in weekly installments during lectures, and online at the end of the corresponding week. On the differences between discretedigital and analog, see. The topic of formal languages and the related topic of finite state automata are not directly addressed in our two references. The present definition of a set may sound very vague. It is increasingly being applied in the practical fields of mathematics and computer science. To introduce the student to the mathematical tools of logic and induction, and to the basic definitions and theorems concerning relations, functions, and sets. Basic building block for types of objects in discrete mathematics. Over the course of five modules, students learn specific skills including binary and modular arithmetic, set notation, methods of. Inverse functions i every bijection from set a to set b also has aninverse function i the inverse of bijection f, written f 1, is the function that assigns to b 2 b a unique element a 2 a such that fa b i observe. The objects that make up a set also known as the sets elements or members can be anything.
The problem sets are challenging but doable if you keep a strong pace. This is a text that covers the standard topics in a sophomorelevel course in discrete mathematics. This is a book about discrete mathematics which also discusses mathematical reasoning and logic. Some people mistakenly refer to the range as the codomain range, but as we will see, that really means the set of all possible outputs even values that the relation. Discrete mathematics is the study of mathematical structures that are countable or otherwise distinct and separable. Being able to do proofs in this setting is the main skill necessary for success in advanced mathematics. Discrete mathematics for computer science some notes jean gallier abstract. Books in this subject area deal with discrete mathematics. Introduction to groups, rings and fields ht and tt 2011 h. A set is typically expressed by curly braces, fgenclosing its elements. Scheinerman, mathematics, a discrete introduction brookscole, pacific grove, ca, 2000. Discrete mathematics department of computer science. Mathematics of discrete structures for computer science.
Discrete mathematics do discrete math books looks boring. Set theory define different types of sets such as, finite and infinite sets, empty set, singleton set, equivalent sets, equal sets, sub sets, proper subsets, supersets, give examples of each kind of set, and solve problems based on them. Discrete structures lecture notes stanford university. Discrete mathematics is the study of mathematical structures that are discrete rather than continuous. Teaches students the mathematical foundations of computer. Discrete structures refers to the study mathematical structures that are individually separate and distinct rather than continuous. M ittra s discrete structures ii cla ss c o n ten tsp refa ce. R tle a x b means r is a set of ordered pairs of the form a,b.
Adding sets and quanti ers to this yields firstorder logic, which is the language of modern mathematics. I admit that i have raised the bar perhaps higher than the average compared to other books on discrete maths. This course will roughly cover the following topics and speci c applications in computer science. Discrete mathematics is the tool of choice in a host of applications, from computers to telephone call routing and from personnel assignments to genetics. Duplicates dont contribute anythi ng new to a set, so remove them. Two sets are equal if and only if they have the same elements.
946 1294 1056 243 91 788 1184 715 630 469 1305 1438 781 972 844 1075 250 458 627 1124 981 1364 587 429 1246 268 670 700 537 1237 1326 1042 47 932 322