Sets and elements set theory is a basis of modern mathematics, and notions of set theory are used in all formal descriptions. Computer science, being a science of the arti cial, has had many of its constructs and ideas inspired by set theory. The story of mathematics, ian stewart describes the situation in this way. The theory of sets was developed by german mathematician georg cantor 18451918. In constructing, we used the informal system of whole numbers as the guide for what.
The fuzzy set follows the infinitevalued logic whereas a crisp set is based on bivalued logic. In non well founded set theories, the foundation axiom of zfc is replaced by axioms implying its negation. Georg cantor this chapter introduces set theory, mathematical in duction, and formalizes the notion of mathematical functions. They are not guaranteed to be comprehensive of the material covered in the course. Nov 02, 2018 the fuzzy set follows the infinitevalued logic whereas a crisp set is based on bivalued logic. A set is a many that allows itself to be thought of as a one. It thus enables us to make predictions about that behavior. By replacing 2 by 1 and considering binary expansions, we obtain a map from the set of such reals onto 0. Set theory, branch of mathematics that deals with the properties of well defined collections of objects, which may or may not be of a mathematical nature, such as numbers or functions. Basic concepts of set theory, functions and relations. Discrete mathematicsset theory wikibooks, open books for. Basically all mathematical concepts, methods, and results admit of representation within axiomatic set theory. Usually we denote sets with uppercase letters, elements with lowercase letters. For example, a set that is identified as the set of even whole numbers between 1 and 11 is a welldefined set because it is possible to identify the exact members of the set.
To summarize, suppose that xis a set to be wellordered, and through zer. A set in mathematics is a collection of well defined and distinct objects, considered as an object in its own right. This lesson looks easy to teach but dont be deceived. In set theory this is done by declaring a universal set. 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 fuzzy set theory is intended to introduce the imprecision and vagueness in order to attempt to model the human brain in artificial intelligence and significance of such theory is increasing day by day in the field of expert systems. Difference between fuzzy set and crisp set with comparison. But even more, set theory is the milieu in which mathematics takes place today. Set theory is indivisible from logic where computer science has its roots. Sets are used to define the concepts of relations and functions. Set theory is the axiomatization of mathematics, and without ac noone seriously doubts its truth, or at least consistency.
When we know what these two sets are and the two conditions are satisfied, we say that f is a well defined function. Set theory \a set is a many that allows itself to be thought of as a one. A definition of a set should describe a set and a unique one and an attempt at a definition may fail because it does not describe any set, or many sets. It has been and is likely to continue to be a a source of fundamental ideas in computer science from theory to practice. Sets are conventionally denoted with capital letters. The set of students in this room over the age of 21. Formal logic is needed to state the axioms of set theory precisely.
Chapter 4 set theory nanyang technological university. Set is also an acronym for secure electronic transaction. Jul 11, 2014 learn what is a set and also learn how to define or describe a set. Description means just that, words describing what is included in a set. The early development of set theory stanford encyclopedia of. The usual set theory of zermelo and fraenkel is not entirely neglected. 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. The objects that make up a set also known as the elements or members of a set can be anything. In mathematics, a welldefined set clearly indicates what is a member of the set and what is not.
The purpose of this module is to introduce language for talking about sets, and some. If we have a set and an object, it is possible that we do not know whether this object belongs to the set or not, because of our lack of information or knowledge. A well defined set has a universe of objects which are allowed into consideration and any object in the universe is either an element of the set or it is not. Local set theory avoids the use of any global universe but instead is formulated in a manysorted language that has various forms of sorts including, for each sort a powersort. We shall say that a set is a well defined collection of objects. Set theory begins with a fundamental binary relation between an object o and a set a. Constructing the integers department of mathematics and. His contributions to set theory as well as to the study of natural numbers and modular lattices are equally important. A set is described by listing elements separated by commas, or by a characterizing property of its elements, within braces.
We shall say that a set is a welldefined collection of objects. What differentiates a set from any group is that a set is well defined. The theory is less valuable in direct application to ordinary experience than as a basis for precise and adaptable terminology for the definition of complex and sophisticated mathematical concepts. In fuzzy set theory this condition was relaxed by lotfi a. Dedekind was a well respected mathematician during his lifetime. We can think of each set as a single entity in itself, and we can denote it and usually do by means of a single symbol. Set theory for computer science university of cambridge. A concept is a symbolic representation of an actual thing tree, chair, table, computer, distance, etc. Non well founded set theories are variants of axiomatic set theory that allow sets to contain themselves and otherwise violate the rule of well foundedness. When discussing sets, the order of the sets does not refer to their arrangement but to their sizes. The membership criteria for a set must in principle be welldefined, and not vague.
The cantor set cis uncountable5 since there is a oneone correspondence6 between elements of cand those reals in 0. We work in set theory without the axiom of choice, so infinite sums and products of cardinal numbers may not be well defined. These notes were prepared using notes from the course taught by uri avraham, assaf hasson, and of course, matti rubin. Introduction to logic and set theory202014 general course notes december 2, 20 these notes were prepared as an aid to the student. Condition i makes sure that each element in x is related to. In set theory as cantor defined and zermelo and fraenkel axiomatized, an object is either a member of a set or not. When one says that, what one really means is simply that whatever was given as a definition is not a definition. Set theory cardinality the cardinality s of s is the number of elements in s. The components of theory are concepts ideally well defined and principles. Set theory, branch of mathematics that deals with the properties of welldefined collections of objects, which may or may not be of a mathematical nature, such as numbers or functions.
In nonwellfounded set theories, the foundation axiom of zfc is replaced by axioms implying its negation the study of nonwellfounded sets was initiated by dmitry mirimanoff in a series of papers between 1917 and 1920. We characterize the sets of integers that can, consistently with zf, occur as the set of. Set theory set up in extensional intuitionistic type theory via setoids is structural. Sets are conceived as objects that have elements, and are related to each other by functions or relations. He first encountered sets while working on problems on. The theory is less valuable in direct application to ordinary experience than as a basis for precise and adaptable terminology for the definition of complex and. 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. Developed at the end of the 19th century, set theory is now a ubiquitous part of mathematics, and can be used as a foundation from which nearly all of. I offer no definition of what a set is beyond the intuitive notion described above. For example, suppose that committee a, consisting of the 5 members jones, blanshard, nelson, smith, and hixon. In this video we discussing how only a well defined collection of objects are called set.
In retrospect zermelos argument for his wellordering theorem proved to be pivotal for the development of set theory. Which set is considered to be well defined in set theory. The theory of the hereditarilyfinite sets, namely those finite sets whose. In fact, his 1900 paper on modular lattices is considered the rst publication in a relatively new branch of mathematics called lattice theory. It was important to free set theory of these paradoxes because nearly all of mathematics was being redefined in terms of set theory. This alone assures the subject of a place prominent in human culture. The reason is that the phrase well defined is not very well defined. Well defined meaning in the cambridge english dictionary. This concept formalizes the idea of grouping objects together and viewing them as a single entity. In set theory, it does not matter how the members of the set are arranged. Therefore, 2, 3, 4, 6, 8, 10 is identical to 10, 2, 4, 8, 6, 3.
Dedekind was a wellrespected mathematician during his lifetime. For example, the degree of membership of a person in the set of tall people is more flexible than a simple yes or no. The set is x is called the domain and denoted by domf, and the set y is called the codomain and denoted by codf. Complex issues arise in set theory more than any other area of pure mathematics. The study of geometry, sequences, probability, etc. Discrete mathematicsset theory wikibooks, open books.
For example, set m is the set of months that start with the letter j. Set theory is one of the greatest achievements of modern mathematics. Pure set theory deals exclusively with sets, so the only sets under consideration are those whose members are also sets. A set is a well defined collection of distinct objects. Discrete mathematicsset theoryexercises wikibooks, open. 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 1831 1916. We will start with, a system that we have already carefully built from set theory. The concept of a set appears in all branches of mathematics.
Roster form lists all of the elements in the set within braces element 1, element 2. The axiom of pair, the axiom of union, and the axiom of. Theory explains how some aspect of human behavior or performance is organized. It does not affect the welldefined nature of a set. In an attempt to avoid these paradoxes, set theory was axiomatized based on firstorder logic, and thus axiomatic set theory was born. Then by the axiom schema of comprehension, there is a set bdf x2vw g.
Posttonal theory set theory, page 1 of 2 set theory a set may be viewed as any welldefined collection of objects. From this simple beginning, an increasingly complex and useful. Sets are one of the most fundamental concepts in mathematics. Set theory is the mathematical theory of well determined collections, called sets, of objects that are called members, or elements, of the set.
Nonwellfounded set theories are variants of axiomatic set theory that allow sets to contain themselves and otherwise violate the rule of wellfoundedness. An infinite set has an unlimited number of members. For example, a set that is identified as the set of even whole numbers between 1 and 11 is a well defined set because it is possible to identify the exact members of the set. In mathematics, a well defined set clearly indicates what is a member of the set and what is not. In the most common structural set theories such as etcs, sets are characterized by the functions between them, i. A set is intuitively defined as any aggregation of objects, called elements, which can be precisely defined in some way or other.
Instead, i am going to show you what can be done with sets. Thus set theory has served quite a unique role by systematizing modern mathematics, and approaching in a unified form all basic questions about admissible. Hardegree, basic set theory page 4 of 39 39 next, we note that set theory officially postulates that a, a,b, a,b,c, etc. Learn what is a set and also learn how to define or describe a set. Px exist, where a is a given set and px is any property of x. For example, zfc has a comprehension axiom asserting that sets of the form xa. We consider the special case of sums or products of countably many copies of the same finite cardinal number n. A welldefined set has a universe of objects which are allowed into consideration and any object. Intuitively, it is any welldefined english assertion about the variable x. A structural set theory is a set theory which describes structural mathematics, and only structural mathematics. Bread a union b or the union of a and bis defined as the set that consists of all elements belonging to either set a or set b or both. This book is dedicated to prabir roy, louis mcauley. An introduction to independence proofs by kenneth kunen, naive set theory by paul r. Although elementary set theory is wellknown and straightforward, the modern subject, axiomatic set theory, is both conceptually more di.