Georg cantor this chapter introduces set theory, mathematical induction, and formalizes the notion of mathematical functions. Two sets are equal if and only if they have the same elements. A set is an unordered collection of different elements. A set may also be thought of as grouping together of single objects into a whole.
In this course you will learn the important fundamentals of discrete math set theory, relations, functions and mathematical induction with the help of 6. The strong tradition, universality and neutrality of set theory make it rm common ground on which to provide uni cation between seemingly disparate areas and notations of computer science. But even more, set theory is the milieu in which mathematics takes place today. From this simple beginning, an increasingly complex and useful. The order of the elements in a set doesnt contribute. Set symbols of set theory and probability with name and definition. The aim of this book is not to cover discrete mathematics in depth it should be clear. This is a course on discrete mathematics as used in computer science.
A book of set theory, first published by dover publications, inc. Issues about data structures used to represent sets and the computational cost of set operations. The objects in a set s are called the members of s. This alone assures the subject of a place prominent in human culture. Set theory has its own notations and symbols that can seem unusual for many. B for example, if s is the set of all juices in the supermarket, and t.
This chapter will be devoted to understanding set theory, relations, functions. The objects in a set are called the elements, or members, of the set. Basic concepts of set theory, functions and relations. Nearly all areas of research be it mathematics, computer science, actuarial science, data science, or even engineering use set theory in one way or the other. In this section, we define a number of operations on sets. I cover all of the important topics thoroughly at a university level with lecture videos, example videos, additional problems, and sample exams with unique and challenging questions that will help you identify your weak points and master the material. Discrete mathsets, relations, functions and math induction.
A beautiful exposition of discrete probability can be found in chapter 8 of concrete mathematics, by graham, knuth, and patashnik 1. This course covers the mathematical topics most directly related to computer science. The notion of set is taken as undefined, primitive, or basic, so we dont try to define what a set is. Number theory athe math behind the rsa crypto system. The set is a welldefined collection of definite objects of perception or thought and the georg cantor is the father of set theory. Translate natural language statements to and from formal propositional logic. Elements of a set can be just about anything from real physical objects to abstract mathematical objects. 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.
Among the fields covered by discrete mathematics are graph and hypergraph theory, enumeration, coding theory, block designs, the combinatorics of partially ordered sets, extremal set theory, matroid theory, algebraic combinatorics, discrete geometry, matrices, and. This chapter develops the basic theory of sets and then explores its connection with combinatorics adding and multiplying. Sets and elements set theory is a basis of modern mathematics, and notions of set theory are used in all formal descriptions. Lecture notes glynn winskel c 2005, 2006 glynn winskel june 26, 2006. Pdf on jan 1, 2010, glynn winskel and others published discrete mathematics ii. This book has been reprinted with the cooperation of. Topics include logic, set theory, number theory, induction, recursion, counting techniques, and graph theory. Problems on discrete mathematics1 ltex at january 11, 2007. Georg cantor in the previous chapters, we have often encountered sets, for example, prime numbers form a set, domains in predicate logic form sets as well. Georg cantor this chapter introduces set theory, mathematical in duction, and formalizes the notion of mathematical functions. Set theory is an mathemati cal language to talk about collections. The notion of set is taken as undefined, primitive, or basic, so we dont try to define what a set is, but we can give an informal description, describe. The universal set is the set that contains everything concerned, usually denoted as u.
I offer no definition of what a set is beyond the intuitive notion described above. In this chapter, we will cover the different aspects of set theory. Today we introduce set theory, elements, and how to build sets. A set may be defined by a membership rule formula or by listing its. This is a book about discrete mathematics which also discusses mathematical rea. Since spring 20, the book has been used as the primary textbook or a supplemental resource at more than 75 colleges and universities around the world. To indicate that an object x is a member of a set a one writes x. This video is an updated version of the original video released over two years ago. Discrete mathematics uses a range of techniques, some of which is seldom found in its continuous counterpart. Duplicates dont contribute anythi ng new to a set, so remove them. Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous.
Set theory basic building block for types of objects in discrete mathematics. Set theory, relations and functions form an integral part of discrete math. A set may also be thought of as grouping together of single objects into. B for example, if s is the set of all juices in the supermarket, and t is the set of all. The mathematics in these applications is collectively called discrete mathematics. Discrete mathematicsset theoryexercises wikibooks, open. An introduction to set theory department of mathematics. The inverse element denoted by i of a set s is an element such that a. Set theory forms the basis of several other fields of study like counting theory, relations, graph theory and finite state machines. Set theory if we are interested in elements of a set a that are not contained in a set b, we can write this set as a.
In naive set theory, a set is a collection of objects called members or elements that is regarded as being a single object. Discrete mathematics set theory cardinality how to compute the cardinality of a set. When expressed in a mathematical context, the word statement is viewed in a. These will be the only primitive concepts in our system. Instead, i am going to show you what can be done with sets. Discrete mathematics, set theory, and logic free math help. Proof involving cartesian products, complements, and universal set discrete math. Introduction to discrete mathematics stanford ai lab. A set is typically determined by its distinct elements, or members, by which we mean. Basic set theory a set is a many that allows itself to be thought of as a one. Most of the notation we use below is standard, although some might be a little. Excellent discussion of group theory applicationscoding. Thanks for contributing an answer to mathematics stack exchange. Basic set theory members of the collection comprising the set are also referred to as elements of the set.
Aug 11, 2018 in this article, we will learn about the introduction of sets and the different types of set which is used in discrete mathematics. Set theory tutorial problems, formulas, examples mba. The text covers the mathematical concepts that students will encounter in many disciplines such as computer science, engineering, business, and the sciences. This book has been reprinted with the cooperation of kyung moon publishers, south korea. Discrete mathematicsset theory wikibooks, open books for. The aim of this book is not to cover discrete mathematics in. Comprehensive coverage of graph theory and combinatorics. Discrete mathematics, set theory, and logic discrete math, graph theory, combinatorics, recurrence relations difference equations, set theory, gene. Set theory and types of set in discrete mathematics.
Set theory examples of set representation of a set famous sets in math equal sets equivalent sets empty set or null set singleton set or unit set universal set slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. Set theory \a set is a many that allows itself to be thought of as a one. The order of a group g is the number of elements in g and. This book is designed for a one semester course in discrete mathematics for sophomore or junior level students. The basic set operations union, intersection and complement on subsets of a fixed. For those of you new to abstract mathematics elementary does not mean simple though much of the material. An important feature of a set is that its elements are \distinct or \uniquely identi able. Besides reading the book, students are strongly encouraged to do all the. For example, the set 2,4,17,23 is the same as the set 17,4,23,2. Sets and set operations cs 441 discrete mathematics for cs m. The following notations will be followed throughout the book. A set can be written explicitly by listing its elements using set bracket. In general, the context of the problem determines u. This book contains a judicious mix of concepts and solved examples that make it ideal for the beginners taking the discrete mathematics course.
They are the fundamental building blocks of discrete math and are highly significant in todays world. Many different systems of axioms have been proposed. International journal of discrete mathematics science. So, a group holds four properties simultaneously i closure, ii associative, iii identity element, iv inverse element. Ab fx 2u jx 2a or x 2bg 2 the intersection of a and b, denoted a \b, is the set of all elements that are both in a and in b. This course will roughly cover the following topics and speci c applications in computer science. Discrete here is used as the opposite of continuous. In this article, we will learn about the introduction of sets and the different types of set which is used in discrete mathematics. A set is an unordered collection of distinct objects. But avoid asking for help, clarification, or responding to other answers. Few full derivations of complex mathematical theorems from set theory. Set theory is likely to be around long after most presentday programming languages have faded from memory. In this tutorial, we look at some solved examples to understand how set theory works and the kind of problems it can be used to solve. 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.
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