This unit also covers subsets, the universal set, set builder notation, complement, intersection and union. In particular we do not assume we are working inside any. The study of geometry, sequences, probability, etc. Set theory is a basis of modern mathematics, and notions of set theory are used in all formal descriptions. Georg cantor this chapter introduces set theory, mathematical in duction, and formalizes the notion of mathematical functions. In this case we also say that acontains b, and we write b. Nb note bene it is almost never necessary in a mathematical proof to remember that a function is literally a set of ordered pairs. A pair f, a is called a soft set over u, where f is a mapping given by f. Set theory tutorial problems, formulas, examples mba. Nafisha tasnim zaman b1506150 shamin yeaser rahman b15061 sarker humaira mostareen b1506014 farjana islam mim b1506182. This alone assures the subject of a place prominent in human culture.
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. Instead, i am going to show you what can be done with sets. A pair f,e is called a soft set over u if and only if f is a mapping ore. The present book covers each of these areas, giving the reader an understanding of the ideas involved. Georg cantor this chapter introduces set theory, mathematical induction, and formalizes the notion of mathematical functions. In other words, a soft set over u is parametrized family of subsets of the universe c. Set theory sets a set is a collection of objects, called its elements. A set can be represented by listing its elements between braces. The concept of set serves as a fundamental part of the present day mathematics. It can be thought of as the mathematical abstraction of the process of making a geographical map.
My aim has been to make this material accessible to a mathematician whose particular. Partially ordered sets a partially ordered set p hp. The purpose of this module is to introduce language for talking about sets, and some. The inversion of a pitch class is the pitch class that is the same number of half steps away from c, but in the opposite direction. 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. Sets are used to define the concepts of relations and functions. The student who gets interested in set theory for its own sake should know, however, that there is much more to the subject than there is in this book. Complex issues arise in set theory more than any other area of pure mathematics. As an illustration, let us consider the following examples. 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. But even more, set theory is the milieu in which mathematics takes place today. This chapter introduces set theory, mathematical in duction, and.
These notes are an introduction to set theory and topology. Sets, functions, relations northwestern university. Set theory has its own notations and symbols that can seem unusual for many. 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. Abstractthe soft set theory offers a general mathematical tool for dealing with uncertain, fuzzy, not clearly. A revised and corrected republication of set theory, originally published in 1971 by addisonwesley publishing company, reading, massachusetts. The elements of a mathematical theory may have some properties, and. Introduction to set theory a solution manual forhrbacek and jech1999 jianfei shen school of economics, the university of new south wales sydney, australia. One of the most beautiful sources of set theoretic wisdom is still hausdorffs set theory. 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 theory 202014 general course notes december 2, 20 these notes were prepared as an aid to the student. Chapter 4 set theory \a set is a many that allows itself to be thought of as a one. Let us consider a few examples of entities we may consider to be sets.
Since strategies for these games can be coded by real numbers, the natural inner model to consider is lr, the smallest model of zermelofraenkel set theory containing the reals and the ordinals. Inversional symmetry this is a property of a set which is able to map onto itself through inversion at some value of n. Nafisha tasnim zaman b1506150 shamin yeaser rahman b15061 sarker humaira. It is designed for a onesemester course in set theory at the advanced undergraduate or beginning. Yz, if f is injective and g is surjective, then g f is surjective. Sets and elements set theory is a basis of modern mathematics, and notions of set theory are used in all formal descriptions. Philosophy of set theory university of california, irvine. 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 important properties of sets, and give examples. Definition 1 a function or a mapping from a to b, denoted by f. The integers are the set of whole numbers, both pos itive and. This book is dedicated to prabir roy, louis mcauley, jonathan. In the example above, the collection of all the possible elements in a is known as the domain. Notes on descriptive set theory and applications to banach.
Although it is customary to use small and capital letters as in x. The progress in past quarter century has blurred the distinction between these areas. Now we will turn to the set mapping reflection principle mrp. The last three examples suggest that a set with elements has 2 subsets. Set theory is indivisible from logic where computer science has its roots. On the right, the collection of all possible outputs also known as range in. They are the result of teaching a twosemester course sequence on these topics for many years at washington university in st.
This instructional unit on sets provide a stepbystep introduction to sets and set theory. Of course, one could aim the book at those who already know all the prerequisites, but chances are that these few potential readers already know descriptive set theory. In mathematics, the notion of a set is a primitive notion. Rethinking set theory tom leinster as mathematicians, we often read a nice new proof of a known theorem, enjoy the di erent approach, but continue to derive our internal understanding from the method we originally learned. I wrote it in the rm belief that set theory is good not just for set theorists, but for many mathematicians, and that the earlier a student sees the particular point of view that we call modern set theory, the better. Set theory empty set the set that contains no element is called the empty set or null set. It is natural for us to classify items into groups, or sets, and consider how those sets overlap with each other. So, before we get started on discussing set theory at all, we will start with a very brief history. Lance small, uc san diego this book is based on notes from a course on set theory and metric spaces taught by edwin spanier, and also incorporates with his permission numerous exercises from those notes. I offer no definition of what a set is beyond the intuitive notion described above. An art collector might own a collection of paintings, while a music lover might keep a collection of cds.
B for example, if s is the set of all juices in the supermarket, and t is the set of all. It has been and is likely to continue to be a a source of fundamental ideas in computer science from theory to practice. Let us see the different types of symbols used in mathematics set theory with its meaning and examples. This accessible approach to set theory for upperlevel undergraduates poses rigorous but simple arguments. For any sentential assignment fthere is a function fmapping the set. They are not guaranteed to be comprehensive of the material covered in the course. To understand that, it will help to think a little bit about the history and mythology of mathematics. For those of you new to abstract mathematics elementary does not mean simple though much of the material. Set theory a mathematical model that we will use often is that of. A little set theory never hurt anybody matthew saltzman department of mathematical sciences clemson university draft. Typically the students were advanced undergraduate mathematics majors, a few beginning graduate students in mathematics, and some graduate students from other areas that included economics and. The mathematical theory of sets is both a foundation in some sense for classical mathematics and a branch of mathematics in its own right.
Sets are welldetermined collections that are completely characterized by their elements. In the sense of a function, a map is often associated with some sort. The set a is called the domain of f, and the set b is called the codomain of f. Basic concepts of set theory, functions and relations. An introduction to independence proofs by kenneth kunen, naive set theory by paul r. Maps may either be functions or morphisms, though the terms share some overlap. An introduction to set theory and topology open scholarship. A set is a collection of objects, called elements of the set.
I will state and give supporting evidence for several theses. A function in set theory world is simply a mapping of some or all elements from set a to some or all elements in set b. The basic relation in set theory is that of elementhood, or membership. Both its foundational role and its particular mathematical features the centrality of axiomatization and the prevalence of. In set theory, inversion means the same number of half steps in the opposite direction. Fe may be considered as the set of capproximate elements of the soft set f. A basic grounding of sentential logic, rstorder logic and axiomatic set theory is assumed, such as that found in 6 or 12, as is an elementary knowledge of group theory and general topology. A set is an unordered collection of things not counting multiplicities, its elements. August 19, 2019 1 introduction the fundamental ideas of set theory and the algebra of sets are probably the most important concepts across all areas of mathematics other than the algebra of real numbers. Basic definitions and notation, types of sets, equality, and venn diagrams are presented. We write x2ato mean that xis an element of a set a, we also say that xbelongs to aor that xis in a. Notes on axioms of set theory well ordering and ordinal numbers123 1. Thus, two sets are equal if and only if they have exactly the same elements. May 05, 2015 applications of set theory in economical problem 1.
Computer science, being a science of the arti cial, has had many of its constructs and ideas inspired by set theory. To understand the philosophical significance of set theory, it will help to have some sense of why set theory arose at all. In contrast to naive set theory, the attitude adopted in an axiomatic development of set theory is that it is not necessary to know what the things are that are called sets or what the relation of membership means. Basic set theory daniel murfet may 16, 2006 throughout we work with the foundation provided by standard zfc fct,section 3. Ling 310, adapted from umass ling 409, partee lecture notes march 1, 2006 p. If aand bare sets, we say that bis a subset of aif every element of b is an element of a. Today this concept is being used in almost every branch of mathematics. 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. This means that 1,2,3 is a set but 1,1,3 is not because 1 appears twice in the second collection. Compare this to the concept of modes of limited transposition found in messiaens theory. This simplification has no harmful effects, and has the merit of reducing the number of primitive notions and axioms of set theory. In mathematics, a mapping, sometimes shortened as map, is a general function between two mathematical objects or structures. The axiom of choice, the lemma of zorn and the hausdor maximal principle 140 appendix.
The reader is assumed to have familiarity with set theory at the level of kunens. 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. Set theory is the axiomatization of mathematics, and without ac noone seriously doubts its truth, or at least consistency. Of sole concern are the properties assumed about sets and the membership relation. This book is dedicated to prabir roy, louis mcauley, jonathan seldin, anil nerode, and maurice bo a, my teachers, and to w. This paper aims to change drastically the way mathematicians think.
Set theory for computer science university of cambridge. Introduction in this paper my main goal is to elaborate on some ideas from a talk which i gave at the workshop on \ set theory and the philosophy of mathematics held at the university of pennsylvania in october 2010. Although elementary set theory is wellknown and straightforward, the modern subject, axiomatic set theory, is both conceptually more di. Basic set theory a set is a many that allows itself to be thought of as a one. Set theory has experienced a rapid development in recent years, with major advances in forcing, inner models, large cardinals and descriptive set theory. The story of mathematics, ian stewart describes the situation in this way. There are other conceptions of set, but although they have genuine mathematical interest they are not our concern here. Set symbols of set theory and probability with name and definition. Set theory basic set theory stanford encyclopedia of. Measure theory is not needed unless we discuss the probability of two types of events. 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. Set theory 1 set theory one of the main objectives of a statistician is to draw conclusions about a population of objects by conducting an experiment. Likewise, one can say that set has fewer than or the same number of elements as set, if there is an injection from to. We can use these sets understand relationships between groups, and to analyze survey data.
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