The notions of set theory, and the ideas behind many of the proofs, present. Instead, i am going to show you what can be done with sets. The concept of set serves as a fundamental part of the present day mathematics. Computer science, being a science of the arti cial, has had many of its constructs and ideas inspired by set theory. There are other conceptions of set, but although they have genuine mathematical interest they are not our concern here. Proofs of settheoretic relations and equalities among sets are among the simplest types of proofs and therefore present an excellent opportunity to familiarize. This chapter introduces set theory, mathematical in.
Even though proofs that invoke the axiom of choice are widely viewed as being ac. 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. A solid foundation on sets is provided for students of all ages. Basic set theory a set is a many that allows itself to be thought of as a one. Discrete mathematicsset theorypage 2 wikibooks, open. No one shall expel us from the paradise which cantor has created for us. Much of the concepts in probability are derived from the consequences of set theory. Measure theory is not needed unless we discuss the probability of two types of events. Review of set theory and probability space i set a set is a collection of objects.
For those of you new to abstract mathematics elementary does not mean simple though much of the material. The foundational rules of set theory the laws listed below can be described as the foundational rules of set theory. Today this concept is being used in almost every branch of mathematics. This unit also covers subsets, the universal set, set builder notation, complement, intersection and union. This unit also covers subsets, the universal set, setbuilder notation, complement, intersection and union. When we partition a set, we break it down into mutually exclusive and exhaustive regions, i. Selecting the material for presentation in this book often came down to deciding how much detail should be provided when explaining. They are not guaranteed to be comprehensive of the material covered in the course. Sets are used to define the concepts of relations and functions. Proof theory was created early in the 20th century by david hilbert to prove the consistency of the ordinary methods of reasoning used in mathematics in arithmetic number theory, analysis and set theory.
Here are some basic subset proofs about set operations. An introduction to independence proofs by kenneth kunen, naive set theory by paul r. Philosophy of set theory university of california, irvine. If aand bare sets, we say that bis a subset of aif every element of b is an element. These notes were prepared using notes from the course taught by uri avraham, assaf hasson, and of course, matti rubin. 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. 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. 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. Complex issues arise in set theory more than any other area of pure mathematics. I felt really bad because i felt that i have only written a bunch of garbage just to restate the obvious rather than proving anything. Let us consider a few examples of entities we may consider to be sets.
Set theory is indivisible from logic where computer science has its roots. This handout is meant primarily for those students who are already familiar with most of the subject matter contained within that is, those who have taken a proofs class before. Set theory sets a set is a collection of objects, called its elements. The study of geometry, sequences, probability, etc. Set theory a mathematical model that we will use often is that of. This instructional unit on sets provide a stepbystep introduction to sets and set theory. Mat231 transition to higher math proofs involving sets fall 2014 8 11. An early introduction of set theory and proofs to engineering students can enrich their intuition and ability to solve comprehensive problems. The formal axiomatic system of ordinary set theory zfc. Set theory deals with objects and their collections.
Chapter 1 logic and set theory to criticize mathematics for its abstraction is to miss the point entirely. Chapter 4 set theory \a set is a many that allows itself to be thought of as a one. A set is a subset of set if every member of is a member of. A set with exactly two members is called a doubleton. Settheoretic independence proofs are worth studying because the indepen.
Set theory is the axiomatization of mathematics, and without ac noone seriously doubts its truth, or at least consistency. This means that 1,2,3 is a set but 1,1,3 is not because 1 appears twice in the second collection. Sets we discussed in class how to formally show that one set is a subset of another and how to show two sets are equal. The lord by wisdom founded the earth, by understanding he established the heavens. Georg cantor this chapter introduces set theory, mathematical induction, and formalizes the notion of mathematical functions. In this diagram, the set a the rectangle is partitioned into sets w,x, and y. Introduction to set theory a solution manual forhrbacek and jech1999 jianfei shen school of economics, the university of new south wales sydney, australia.
Set theory empty set the set that contains no element is called the empty set or null set. Ling 310, adapted from umass ling 409, partee lecture notes march 1, 2006 p. A little set theory never hurt anybody matthew saltzman department of mathematical sciences clemson university draft. Expressing the importance of georg cantors set theory in the development of mathematics. In set theory this is done by declaring a universal set. Indeed, one way to state the axioms of probability involves set theory. The mathematical theory of sets is both a foundation in some sense for classical mathematics and a branch of mathematics in its own right. Introduction to logic and set theory 202014 general course notes december 2, 20 these notes were prepared as an aid to the student. In mathematics, the notion of a set is a primitive notion. The venn diagram below should help you get the picture. Constructive set theory is an approach to mathematical constructivism following the program of axiomatic set theory, using the usual firstorder language of classical set theory.
Bob solovay and tony martin in the beginning, aleko kechris, ken kunen and leo harrington a little later. We derive them by going back to the definitions of intersection, union, universal set and empty set, and by considering whether a. Basic set theory daniel murfet may 16, 2006 throughout we work with the foundation provided by standard zfc fct,section 3. In this case we also say that acontains b, and we write b. This book is dedicated to prabir roy, louis mcauley. Although elementary set theory is wellknown and straightforward, the modern subject, axiomatic set theory, is both conceptually more di. 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. Set theory is a basis of modern mathematics, and notions of set theory are used in all formal descriptions.
Set theory and proofs for engineering education asee peer logo. A set theory textbook can cover a vast amount of material depending on the mathematical background of the readers it was designed for. Elements of set theory by sidneyfelder in the words of georgcantor 18451918,the creator of set theory and the modern theory of the in. It has been and is likely to continue to be a a source of fundamental ideas in computer science from theory to practice. 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. Set theory for computer science university of cambridge. This book provides an introduction to relative consistency proofs in axiomatic set theory, and is intended to be used as a text in beginning graduate courses in. It is used as a foundation for many subfields of mathematics. This note is an introduction to the zermelofraenkel set theory with choice zfc. If you concentrate too closely on too limited an application of a mathematical idea, you rob the mathematician of his most important tools. In the areas pertaining to statistics, it is particularly used in probability.
If aand bare sets, we say that bis a subset of aif every element of b is an element of a. Set theorysets wikibooks, open books for an open world. 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. 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. B for example, if s is the set of all juices in the supermarket, and t is the set of all. Free set theory books download ebooks online textbooks. In particular we do not assume we are working inside any. M being the only minimal element on that set, is x the least element as. This book is dedicated to prabir roy, louis mcauley, jonathan seldin, anil nerode, and maurice bo a, my teachers, and to w. After mathematical logic has provided us with the methods of reasoning we start with a very basic theory. This book is dedicated to prabir roy, louis mcauley, jonathan.
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 set theory, inversion means the same number of half steps in the opposite direction. The empty set is a subset of a, hence it is an element of the power set of a. An overview of logic, proofs, set theory, and functions. I simply just cant find any relevance to the question with the notes i was studying. Proof, sets, and logic department of mathematics boise state. Sets and elements set theory is a basis of modern mathematics, and notions of set theory are used in all formal descriptions. For any set a, the empty set is an element of the power set of a. It is designed for a onesemester course in set theory at the advanced undergraduate or beginning. We cant think of examples of the use of intransitive verbs. Nb note bene it is almost never necessary in a mathematical proof to remember that a function is literally a set of ordered pairs. Georg cantor this chapter introduces set theory, mathematical in duction, and formalizes the notion of mathematical functions. Both its foundational role and its particular mathematical features the centrality of axiomatization and the prevalence of. I offer no definition of what a set is beyond the intuitive notion described above.
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