Set theory lecture notes by jonathan pila file type. 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. I suppose i might as well mention my own lecture notes, available online here scroll down to set theory. Professor zwiebach has not thoroughly proofread these notes but believes they will be useful to people that want to know what goes on during lecture and what material gets. Averages of arithmetic functions lecture 3, 1012014 10 1. Usually we denote sets with uppercase letters, elements with lowercase letters.
A set is described by listing elements separated by commas, or by a characterizing property of its elements, within braces. Contei\t lecture 1 language, axioms, and elementary constructions of set theory lecture 2 elementary consequences of axioms lecture 3 cartesian products, relations lecture 4 order relations lecture 5 functions lecture 6 natural numbers lecture 7 equipollence and cardinal numbers lecture 8 hierarchy of cardinal numbers lecture 9 arithmetic of cardinal numbers. Mathematics lecture notes on elementary topology and geometry undergraduate texts in mathematics material type book language english title lecture notes on elementary topology and geometry undergraduate texts in mathematics. On the other hand, having iii proven, enabled us to construct a proof of iv the. They are the most basic kind of mathematical structure. This chapter introduces set theory, mathematical in duction. These are lecture notes from an introduction to axiomatic set theory taught by scott weinstein in the spring of 2009, typed up by me and vilhelm sjoberg.
So greentao theorem clearly supports erdos conjecture. Set theory an introduction to independence proofs studies. A polish metric space is universal if it contains an isometric copy of any other polish metric space equivalently, of every countable metric space. A set is pure if all of its members are sets, all members of its members are sets, and so on.
If you continue browsing the site, you agree to the use of cookies on this website. Elementary number theory lecture notes with tests for students of specialties informatics and applied mathematics approved at meeting of department of applied and computational mathematics as the abstract of lecture notes for the discipline elementary number theory record 10 from 19. Lecture notes on elementary topology and geometry undergraduate texts in mathematics author. Note that he refers to functions as transformations and to sets as systems. Lecture 1 language, axioms, and elementary constructions of set theory lecture 2 elementary consequences of axioms lecture 3 cartesian products, relations lecture 4 order relations lecture 5 functions lecture 6 natural numbers lecture 7 equipollence and cardinal numbers lecture 8 hierarchy of cardinal numbers lecture 9 arithmetic of cardinal. Set theory lecture notes, spring 2009 hendrix college. The usual set theory of zermelo and fraenkel is not entirely neglected. The objects of a set are called elements or members. Set theory begins with a fundamental binary relation between an object o and a set a. What are good booksother readings for elementary set theory. The theory intends to capture in an invariant way the notion of a constant abstract set. An elementary theory of the category of sets 7 an elementary theory of the category of sets long version f. Although elementary set theory is wellknown and straightforward, the modern subject. Analytic number theory lecture notes 7 note that none of the factors in the right hand side of 4 vanishes, since jp sj p 1.
Balasubramanian the institute of mathematical sciences, chennai. Hence, it turns out to be useful to introduce a general notion, that of a ring. Elementary number theory lecture notes lior silberman. May 05, 2015 applications of set theory in the field of economics slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. Basic definitions a b x, such that x a and x b a b a b x b such that x a a b x. They describe all possible intersections between several sets. Elementary set theory is an extension of the lecture notes for the course fundamental concepts of mathematics given each year to firstyear undergraduate. I make no guarantees whatsoever as to their completeness or correctness. Using s for the gcd and s for the lcm is, in fact, the traditional notation. For example, the set containing only the empty set is a nonempty pure set. Lecture notes on elementary logic and set theory by jeanmarc. The boolean algebra allows us to treat logical statements and propositions algebraically as we used to do with other mathematical objects such as numbers or. And now we finally arrive at the two main theorems of the course.
Sets and elements set theory is a basis of modern mathematics, and notions of set theory are used in all formal descriptions. The purpose of this module is to introduce language for talking about sets, and some. So sets can consist of elements of various natures. Sets are usually denoted by uppercase letters like a,b,c. I have looked through other books in the rather new ams series student mathematical library and as a collection they seem to be very nice.
These lecture notes were prepared in latex by bryan owens, an mit student. These notes were prepared using notes from the course taught by uri avraham, assaf hasson, and of course, matti rubin. Up to minor adaptations and corrections, it is a translation of an earlier french lecture note for a course given in bordeaux in 20012002. In this project we will learn elementary set theory from the original historical sources by two key. Book, english, lecture notes on elementary topology and geometry undergraduate texts in mathematics keywords. Language of set theory and the first axioms, the power set axiom, the axiom of infinity and the natural numbers, recursion on the natural numbers, arithmetic on the natural numbers, the axioms of replacement and foundation, cardinality, finite. These notes for a graduate course in set theory are on their way to be. There is a study of somewhat more advanced topics in set theory at the. In this set of notes, we begin our foray into learning theory.
As such, it is the prototypical structural set theory. Studies in logic and the foundations of mathematics, volume 102. Urysohn constructed such a space with a random construction which predates the random graph. Examples of structures the language of first order logic is interpreted in mathematical structures, like the following. 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. An introduction to set theory university of toronto. Elementary number theory a revision by jim hefferon, st michaels college, 2003dec of notes by w. In mathematics, the notion of a set is a primitive notion. Book, english, lecture notes on elementary topology and geometry undergraduate texts in mathematics created date. Proposed shortly after etcc in it is also the paradigm for a categorical foundation of mathematics. But even more, set theory is the milieu in which mathematics takes place today. S in etcs has no properties except that it is an element of s and is distinct from any other elements of s. This book is dedicated to prabir roy, louis mcauley, jonathan.
The book contains exercises in most chapters, in line with its super cial character of being an elementary set theory text, but no representation as. Basic concepts of set theory, functions and relations. Lecture notes on elementary topology and geometry undergraduate texts in mathematics details category. Roman v belavkin bis3226 contents 1 sets and operations on sets 1 2 correspondences and mappings between sets 3 the bishop gave monkey the banana. Thenatural number structure in etcs is a triad of a set n, a selected element 0. Whenever i have one of those moments where a definition just doesnt make sense, or a theorem just doesnt seem right its invariably a set of lecture notes that sets everything straight for me. 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. Introduction to logic and set theory 202014 general course notes december 2, 20 these notes were prepared as an aid to the student.
For those of you new to abstract mathematics elementary does not mean simple though much of the material. The terms set, collection, and family are synonymous. Given any two events or sets a and b, we have the following elementary set operations. Weiss university of toronto, 2008 these notes for a graduate course in set theory cover the axioms of set theory, the natural numbers, the ordinal numbers, relations and orderings, cardinality, the real numbers, the universe, reflection, elementary submodels, and constructibility. Georg cantor this chapter introduces set theory, mathematical induction, and formalizes the notion of mathematical functions. Lecture notes string theory for undergraduates physics. Introduction to string theory 3 tures on string theory were supposed to give a pedagogical introduction and to prepare for later lectures on branes worlds, large extra dimensions, the adscft correspondence and black holes. An introduction to independence proofs offers an introduction to relative consistency proofs in axiomatic set theory, including combinatorics, sets, trees, and forcing. 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. This alone assures the subject of a place prominent in human culture.
They are not guaranteed to be comprehensive of the material covered in the course. An elementary theory of the category of sets 3 the sets of etcs are abstract structures in exactly this sense. The book first tackles the foundations of set theory and infinitary combinatorics. The notion of set is taken as undefined, primitive, or basic, so we dont try to define what a set is. Ling 310, adapted from umass ling 409, partee lecture notes march 1, 2006 p. Whereas one of the principal concerns of the latter theory is the deconposition of numbers into prime factors, addi. This section provides lecture notes for the course. Discrete mathematics for computer science 2 a b x, if x a then x b a b definition.
I think it is fair to say that these are a digest version of kaplanskys book, even though they were for the most part not written with that book in hand. Apart from being interesting and enlightening in its own right, this discussion will also help us hone our intuitions and derive rules of thumb about how to best apply learning algorithms in di. Nb note bene it is almost never necessary in a mathematical proof to. These are lecture notes for a class at eth in the spring semester 2010, containing an essentially standard course of measure theory and integration theory with a smattering of probability. It is roughly parallel in structure to halmoss classic naive set theory, though more topics have been added. Discrete mathematics for computer science 5 venn diagrams for sets are similar to truth tables in propositional logic. Basic set theory a set is a many that allows itself to be thought of as a one. Set of primes contains arbitrary long arithmetic progression. Lecture 1 introduction in additive number theory we make reference to facts about addition in 1 contradistinction to multiplicative number theory, the foundations of which were laid by euclid at about 300 b. These lecture notes mostly follow the lectures, but aim to extend them in two ways.
These lecture notes cover the onesemester course introduction to number theory uvod do teorie. Elements of set theory by sidneyfelder in the words of georgcantor 18451918,the creator of set theory and the modern theory of the in. We sometimes write s for the gcd of the set s rather than for the ideal generated by s. After dirichlets death, dedekind edited dirichlets lectures on. The elementary theory of the category of sets, or etcs for short, is an axiomatic formulation of set theory in a categorytheoretic spirit. Ling 409, partee lecture notes, lecture 1 september 7, 2005 p. Introducing sets sets are the rst of the three languages of mathematics. Show that a set of nelements has exactly 2nsubsets for any nonnegative integer n.
This book is dedicated to prabir roy, louis mcauley. There are many books on elementary number theory, most of. If n 0, then sis the empty set and has only one subset, namely the empty set itself. First, the theory characterizes the category of sets and mappings as an abstract category. Set theory an introduction to independence proofs studies in. He used handwritten notes taken in class by kayla jacobs, an mit graduate, in producing them.
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