Set theory begins with a fundamental binary relation between an object o. It is natural for us to classify items into groups, or sets, and consider how those sets overlap with each other. Module 1 for teachers of primary and secondary mathematics. This set includes all rational numbers, together with all irrational numbers that is, algebraic numbers that cannot be rewritten as. Instead, we will use what is callednaive set theory. Which one of the following statements is true about.
Sets and elements set theory is a basis of modern mathematics, and notions of set theory are used in all formal descriptions. It is hoped that this will encourage and facilitate further development of the subject. Georg cantor this chapter introduces set theory, mathematical induction, and formalizes the notion of mathematical functions. Let us see the different types of symbols used in mathematics set theory with its meaning and examples. Chapter 4 set theory nanyang technological university. Ling 310, adapted from umass ling 409, partee lecture notes march 1, 2006 p. This accessible approach to set theory for upperlevel undergraduates poses rigorous but simple arguments.
Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects. The set of 4 and 12 is the intersection of sets a and b. In mathematics, a set is a welldefined collection of distinct objects, considered as an object in. Although elementary set theory is wellknown and straightforward, the modern subject, axiomatic set theory, is both conceptually more di.
A venn diagram is used to showvisualize the possible relations among a collection of sets. The improving mathematics education in schools times project. Graph theory, abstract algebra, real analysis, complex analysis, linear algebra, number theory, and the list goes on. The presentation will show you the methods of finding rank of matrix. He first encountered sets while working on problems. This is the set of all elements currently under consideration, and is often symbolized by.
For example, the natural numbers are referred to by the mathematical symbol n, the english words the natural numbers, and the settheoretic notation 1,2,3. These notes were prepared using notes from the course taught by uri avraham, assaf hasson, and of course, matti rubin. Algebraic set theory uses the methods of category theory to study elementary set theory. Chapter 2 the basic concepts of set theory ppt download. Set theory for computer science university of cambridge. Chapter 2 set theory relations and functions ppt ppt. Ppt set theory powerpoint presentation free to download id. For set union and set difference, the two relations involved must be unioncompatible that is, the two relations must have the. He first encountered sets while working on problems on. A single paper, however, founded set theory, in 1874 by georg cantor. Description chapter 2 set theory relations and functions. Chapter 4 set theory \a set is a many that allows itself to be thought of as a one. The axioms of set theory of my title are the axioms of zermelofraenkel set theory, usually thought ofas arisingfromthe endeavourtoaxiomatise the cumulative hierarchy concept of set.
This series of lessons cover the essential concepts of math set theory the basic ways of describing sets, use of set notation, finite sets, infinite sets, empty sets, subsets, universal sets, complement of a set, basic set operations including intersection and union of sets, using venn diagrams and simple applications of sets. Set theory is the branch of mathematical logic that studies sets, which informally are collections of objects. Set theory tutorial problems, formulas, examples mba. Basic concepts of set theory, functions and relations. Set theory is indivisible from logic where computer science has its roots. Set theory, relations, functions set u a set b is a subset of a which is subset of universal set u. Cook liberty university department of mathematics fall 2016. In this lesson we will learn about sets, subsets, venn diagr. Abstract algebra is a relatively modern topic in mathematics. Set theory is not really the only rigorous mathematical language.
This is a book on linear algebra and matrix theory. Set theory, abstract algebra, number theory, real analysis, topology, complex analysis, and linear algebra. 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. Relationship between graph theory and linear algebra powerpoint ppt presentation. There were several threads in the early development of group theory, in modern language loosely. Powerpoint presentations ppt collection for mathematics. 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. Introduction to logic and set theory202014 general course notes december 2, 20 these notes were prepared as an aid to the student. Pure mathematics for beginners consists of a series of lessons in logic, set theory, abstract algebra, number theory, real analysis, topology, complex analysis, and linear algebra. Ppt set theory powerpoint presentation, free download id. Lesson 2 from this series covers the basics of set theory. Preface this book is designed for a one semester course in discrete mathematics for sophomore or junior level students. Such as, sets could be a collection of odd numbers, even numbers, natural numbers, whole numbers, real or complex numbers and all the set of numbers which comes in the number line. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics.
I never had a course that was specifically about set theory in my math education writing my phd thesis atm. If a set doesnt have any elements, it is known as an empty set or null set or void set. Mathematics introduction of set theory a set is an unordered collection of objects, known as elements or members of the set. Sets are used to define the concepts of relations and functions. There cannot be any application because set theory is selfcontradictory and above all is in contradiction with mathematics. Other places in the world also had algebra and developed various aspects of it. In standard introductory classes in algebra, trigonometry, and calculus there is currently very lit tle emphasis on the discipline of proof. The languages of set theory and of mathematical logic were developed together, so that, as a mathematical discipline, set theory is a branch of mathematical logic. The study of geometry, sequences, probability, etc. These will be the only primitive concepts in our system. When expressed in a mathematical context, the word statement is viewed in a. Chapter 6 abstract algebra groups rings field lattics and boolean algebra 6.
But as you say set theory is kind of the fundamentals of math. B for example, if s is the set of all juices in the supermarket, and t is the set of all. Instead, i am going to show you what can be done with sets. For those of you new to abstract mathematics elementary does not mean simple though much of the material. We could say set c is the intersection of a and b, and its this set right over here. Chapter 2 set theory relations and functions ppt add to favourites. Set theory has its own notations and symbols that can seem unusual for many. But there is one thing that all of these share in common. Georg cantor, one of the founders of set theory, gave the following definition of a set at the. And we could even, if we want to, we could even label this as a new set.
While it is self contained, it will work best for those who have already had some exposure to linear algebra. Introduction to pure mathematics for advanced high school students consists of a series of lessons in logic, set theory, abstract algebra, number theory, real analysis, topology, complex analysis, and linear algebra. In abstract algebra, the inverse of an element a is usually written a1. 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 courses where we needed some set theory they just introduced it when needed. Set theory and algebra in computer science a gentle introduction to mathematical modeling jos. Set theory is commonly used as a foundational system, although in some areas category theory is thought to be a preferred foundation. Fuzzy set theory and its applications, third edition is a textbook for courses in fuzzy set theory. The theory of sets was developed by german mathematician georg cantor 18451918. Im not really sure what you learn in a course on elementary set theory. Group theory powerpoint group mathematics integer scribd. It can also be used as an introduction to the subject.
I offer no definition of what a set is beyond the intuitive notion described above. Survey of mathematical ideas math 100 chapter 2 john rosson thursday january. The ancient chinese solved systems of simultaneous linear equations and later developed algorithms to nd roots of polynomials of high degree. Ppt introduction to set theory powerpoint presentation. The purpose of this website is to link together current research in algebraic set theory and make it easily available. The axioms of set theory, the natural numbers, the ordinal numbers, relations and orderings, cardinality, there is nothing real about the real numbers, the universe, reflection, elementary submodels and constructibility. For example, m x x is a month of the year and x starts with the letter j this is read, set m is the set of all the elements x such that x is a month of the year and x starts with the.
The relational algebra uses set union, set difference, and cartesian product from set theory, but adds additional constraints to these operators. It is also assumed that the reader has had calculus. It happened that my double major in physics kept me away from the lecture time for the course. Group theory powerpoint free download as powerpoint presentation. Some optional topics require more analysis than this, however. If the set has a nite number of elements, this is called a \ nite group. Sets, elements any well defined list or collection of objects is called a set. The notes are designed to be used in conjunction with a set of online homework exercises which help the students read the lecture notes and learn basic linear algebra skills.
Complex issues arise in set theory more than any other area of pure mathematics. Usually we denote sets with uppercase letters, elements with lowercase letters. And the number, i guess, 10 and 3 is only in b, so were done. There are other conceptions of set, but although they have genuine mathematical interest they are not our concern here. Basic set theory a set is a many that allows itself to be thought of as a one. Nb note bene it is almost never necessary in a mathematical proof to remember that a function is literally a set of ordered pairs. Mathematics introduction of set theory geeksforgeeks. Set theory introduction this chapter treats some of the elementary ideas and concepts of set theory which are necessary for a modern introduction to probability theory. An introduction to elementary set theory guram bezhanishvili and eachan landreth 1 introduction in this project we will learn elementary set theory from the original historical sources by two key gures in the development of set theory, georg cantor 18451918 and richard dedekind 18311916. A free powerpoint ppt presentation displayed as a flash slide show on id. Set theory and algebra in computer science a gentle. The following notation is used to show set membership means that x is a member of the set a means that x is not a member of the set a. Set theory, relations, functions a set of three objects. Technically, as we shall see shortly, we can view the language of set theory as a special sublanguage of.
A revised and corrected republication of set theory, originally published in 1971 by addisonwesley publishing company, reading, massachusetts. Here we are not concerned with a formal set of axioms for set theory. Introduction to logic and set theory 202014 general course notes december 2, 20 these notes were prepared as an aid to the student. Our goal is to give the beginning student, with little or no prior exposure to linear algebra, a good grounding in the basic ideas, as well as an appreciation for how they are used in many applications, including data tting, machine learning and arti cial intelligence, to. Set theory algebra cs gate syllabus, paper solution. They are not guaranteed to be comprehensive of the material covered in the course.
Richard mayr university of edinburgh, uk discrete mathematics. Computer science, being a science of the arti cial, has had many of its constructs and ideas inspired by set theory. An element a belong to a set a can be written as a. In fact, when i took this course it was called modern algebra. On a characteristic property of all real algebraic numbers.
Ppt chapter 6 abstract algebra powerpoint presentation. A welldefined collection of objects the set of outstanding people, outstanding is very subjective. 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. It has been and is likely to continue to be a a source of fundamental ideas in computer science from theory to practice. The theory of these and their use in quantum mechanics is a welldeveloped subject, but one we mostly will bypass in favor of the study.
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