Well defined set theory pdf

The components of theory are concepts ideally well defined and principles. Introduction to logic and set theory202014 general course notes december 2, 20 these notes were prepared as an aid to the student. The usual set theory of zermelo and fraenkel is not entirely neglected. He first encountered sets while working on problems on trigonometric series. The axiom of pair, the axiom of union, and the axiom of. A set is a many that allows itself to be thought of as a one.

A set is described by listing elements separated by commas, or by a characterizing property of its elements, within braces. By replacing 2 by 1 and considering binary expansions, we obtain a map from the set of such reals onto 0. 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. From this simple beginning, an increasingly complex and useful. For example, zfc has a comprehension axiom asserting that sets of the form xa. Jul 11, 2014 learn what is a set and also learn how to define or describe a set. 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. The study of geometry, sequences, probability, etc. Set theory is the mathematical theory of well determined collections, called sets, of objects that are called members, or elements, of the set. Set theory \a set is a many that allows itself to be thought of as a one. Set theory set up in extensional intuitionistic type theory via setoids is structural. In set theory, it does not matter how the members of the set are arranged.

Formal logic is needed to state the axioms of set theory precisely. 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. The set of students in this room over the age of 21. This concept formalizes the idea of grouping objects together and viewing them as a single entity. There is a study of somewhat more advanced topics in set theory at the. Posttonal theory set theory, page 1 of 2 set theory a set may be viewed as any welldefined collection of objects. Set is also an acronym for secure electronic transaction. Px exist, where a is a given set and px is any property of x. This alone assures the subject of a place prominent in human culture.

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. It has been and is likely to continue to be a a source of fundamental ideas in computer science from theory to practice. 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. In mathematics, a welldefined set clearly indicates what is a member of the set and what is not. Set theory is one of the greatest achievements of modern mathematics. This lesson looks easy to teach but dont be deceived. A set in mathematics is a collection of well defined and distinct objects, considered as an object in its own right. Discrete mathematicsset theoryexercises wikibooks, open. The set is x is called the domain and denoted by domf, and the set y is called the codomain and denoted by codf.

Set theory is the axiomatization of mathematics, and without ac noone seriously doubts its truth, or at least consistency. Usually we denote sets with uppercase letters, elements with lowercase letters. A set is a well defined collection of distinct objects. Set theory is indivisible from logic where computer science has its roots. 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. Georg cantor this chapter introduces set theory, mathematical in duction, and formalizes the notion of mathematical functions. A welldefined set has a universe of objects which are allowed into consideration and any object. When discussing sets, the order of the sets does not refer to their arrangement but to their sizes. Pure set theory deals exclusively with sets, so the only sets under consideration are those whose members are also sets. Computer science, being a science of the arti cial, has had many of its constructs and ideas inspired by set theory. It does not affect the welldefined nature of a set.

Constructing the integers department of mathematics and. They are not guaranteed to be comprehensive of the material covered in the course. 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. It thus enables us to make predictions about that behavior. The story of mathematics, ian stewart describes the situation in this way. Although elementary set theory is wellknown and straightforward, the modern subject, axiomatic set theory, is both conceptually more di. 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. 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. In non well founded set theories, the foundation axiom of zfc is replaced by axioms implying its negation. Set theory begins with a fundamental binary relation between an object o and a set a. Instead, i am going to show you what can be done with sets.

Set theory cardinality the cardinality s of s is the number of elements in s. Then by the axiom schema of comprehension, there is a set bdf x2vw g. The cantor set cis uncountable5 since there is a oneone correspondence6 between elements of cand those reals in 0. In fuzzy set theory this condition was relaxed by lotfi a. 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. 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. In set theory as cantor defined and zermelo and fraenkel axiomatized, an object is either a member of a set or not. It was important to free set theory of these paradoxes because nearly all of mathematics was being redefined in terms of set theory. An introduction to independence proofs by kenneth kunen, naive set theory by paul r. 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. A set is intuitively defined as any aggregation of objects, called elements, which can be precisely defined in some way or other. Theory explains how some aspect of human behavior or performance is organized.

A concept is a symbolic representation of an actual thing tree, chair, table, computer, distance, etc. 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. We shall say that a set is a welldefined collection of objects. In set theory this is done by declaring a universal set. Discrete mathematicsset theory wikibooks, open books. Complex issues arise in set theory more than any other area of pure mathematics. The concept of a set appears in all branches of mathematics. We consider the special case of sums or products of countably many copies of the same finite cardinal number n. 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. Set theory for computer science university of cambridge. Nov 02, 2018 the fuzzy set follows the infinitevalued logic whereas a crisp set is based on bivalued logic. Sets are one of the most fundamental concepts in mathematics.

His contributions to set theory as well as to the study of natural numbers and modular lattices are equally important. We characterize the sets of integers that can, consistently with zf, occur as the set of. To summarize, suppose that xis a set to be wellordered, and through zer. Roster form lists all of the elements in the set within braces element 1, element 2. 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.

Well defined meaning in the cambridge english dictionary. Discrete mathematicsset theory wikibooks, open books for. The fuzzy set follows the infinitevalued logic whereas a crisp set is based on bivalued logic. Description means just that, words describing what is included in a set.

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. 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. Chapter 4 set theory nanyang technological university. But even more, set theory is the milieu in which mathematics takes place today.

The theory of the hereditarilyfinite sets, namely those finite sets whose. Sets are conventionally denoted with capital letters. Sets and elements set theory is a basis of modern mathematics, and notions of set theory are used in all formal descriptions. Sets are conceived as objects that have elements, and are related to each other by functions or relations. For example, the degree of membership of a person in the set of tall people is more flexible than a simple yes or no. In mathematics, a well defined set clearly indicates what is a member of the set and what is not. Which set is considered to be well defined in set theory. Learn what is a set and also learn how to define or describe a set. The early development of set theory stanford encyclopedia of. Condition i makes sure that each element in x is related to.

These notes were prepared using notes from the course taught by uri avraham, assaf hasson, and of course, matti rubin. The reason is that the phrase well defined is not very well defined. When we know what these two sets are and the two conditions are satisfied, we say that f is a well defined function. The objects that make up a set also known as the elements or members of a set can be anything. I offer no definition of what a set is beyond the intuitive notion described above. Basically all mathematical concepts, methods, and results admit of representation within axiomatic set theory. In constructing, we used the informal system of whole numbers as the guide for what.

What differentiates a set from any group is that a set is well defined. 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. The purpose of this module is to introduce language for talking about sets, and some. The theory of sets was developed by german mathematician georg cantor 18451918. The membership criteria for a set must in principle be welldefined, and not vague. We work in set theory without the axiom of choice, so infinite sums and products of cardinal numbers may not be well defined. Dedekind was a well respected mathematician during his lifetime. Thus set theory has served quite a unique role by systematizing modern mathematics, and approaching in a unified form all basic questions about admissible. Basic concepts of set theory, functions and relations. We will start with, a system that we have already carefully built from set theory. Sets are used to define the concepts of relations and functions. An infinite set has an unlimited number of members. In an attempt to avoid these paradoxes, set theory was axiomatized based on firstorder logic, and thus axiomatic set theory was born.

In fact, his 1900 paper on modular lattices is considered the rst publication in a relatively new branch of mathematics called lattice theory. In this video we discussing how only a well defined collection of objects are called set. 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. For example, set m is the set of months that start with the letter j. We shall say that a set is a well defined collection of objects. When one says that, what one really means is simply that whatever was given as a definition is not a definition.

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. For example, suppose that committee a, consisting of the 5 members jones, blanshard, nelson, smith, and hixon. Dedekind was a wellrespected mathematician during his lifetime. In retrospect zermelos argument for his wellordering theorem proved to be pivotal for the development of set theory. Zadeh so an object has a degree of membership in a set, a number between 0 and 1. Therefore, 2, 3, 4, 6, 8, 10 is identical to 10, 2, 4, 8, 6, 3.

In the most common structural set theories such as etcs, sets are characterized by the functions between them, i. Intuitively, it is any welldefined english assertion about the variable x. Nonwellfounded set theories are variants of axiomatic set theory that allow sets to contain themselves and otherwise violate the rule of wellfoundedness. He first encountered sets while working on problems on. 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.