It has been and is likely to continue to be a a source of fundamental ideas in computer science from theory to practice. These notes are an introduction to set theory and topology. Although elementary set theory is wellknown and straightforward, the modern subject, axiomatic set theory, is both conceptually more di. How to read music notes quick learn cheat sheets, page 3 note. Lectures notes for an advanced course given in esslli 2010. 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. This paper has appeared in the proceedings of the logic colloquium 2006, lecture notes in.
Basic set theory members of the collection comprising the set are also referred to as elements of the set. In order to eliminate such problems, an axiomatic basis was developed for the theory of sets analogous to that developed for elementary geometry. The idea of placing notes on a set of horizontal lines the staff to show the relationship of various pitches with each other staff notation emerged in western music towards the end of the ninth century. A modern description of what ergodic theory is would be. What appeals to me most about descriptive set theory is that to study it you must reallyunderstandso many things. The idea is that two sets are equivalent if it is possible to pair off members of the first set with members of the second, with. Challenge students to develop speed and accuracy in their note naming with this free worksheet. Set theory and relations set theory a set is well defined class or collection of objects. A final chapter presents an alternative conception of set theory useful in computer science. Of the usual socalled zermelofraenkel axioms with choice, the. The stem on the note b can face either direction b gd g d bf b f e g f d b e. It is designed for a onesemester course in set theory at the advanced undergraduate or beginning. Jech a comprehensive advanced text which goes well beyond the above syllabus.
With the clock ticking, students name ten notes as fast as they can. Basic set theory lx 502 semantics i september 11, 2008 1. Set theory frank stephan semester i, academic year 20092010 set theory deals with the fundamental concepts of sets and functions used everywhere in mathematics. An introduction to set theory and topology by ronald c. Set theory is a basis of modern mathematics, and notions of set theory are used in all formal descriptions. Note you should be familiar with logic, as explained in the previous chapter. The staff paper offers a wide range and is available a convenient option for meeting all kinds of music paper requirements. 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 a mathematical model that we will use often is that of. If updating is not possible, you may view the 2017 edition of the website. Draw the treble clef at the beginning of the staff and then draw the notes indicated.
A proper subset is a subset that is not identical to the original setit contains fewer elements. Applications of set theory slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. It then delves into contemporary set theory, covering such topics as the borel hierarchy and lebesgue measure. They originated as handwritten notes in a course at the university of toronto given by prof. There are other conceptions of set, but although they have genuine mathematical interest they are not our concern here. Draw a treble clef at the beginning of the staff and write the letter name of each note. 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. At just that time, however, several contradictions in socalled naive set theory were discovered. The printable staff paper is a lined paper template for the creation and writing of musical notes. These elements could be numbers, alphabets, variables, etc. Set e6 has as its elements all employee records whose. To view the requested content, install the latest version of one of the following modern web browsers.
Set theory for computer science university of cambridge. They are the result of teaching a twosemester course sequence on these topics for many years at washington university in st. 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. A preliminary version of the second greek edition of notes on set theory.
This book is a mammoth survey that covers everything from the basics of set theory to recent research such as models of set theory involving large cardinals. It can be used by the beginners, musicians, teachers and students for creating and printing staff sheets. B for example, if s is the set of all juices in the supermarket, and t is the set of all. Categorical semantics of constructive set theory homepages of. Set theory as a foundation for mathematical analysis, topology, abstract algebra, and discrete.
These notes were prepared as an aid to the student. The study of geometry, sequences, probability, etc. The proofs of theorems files were prepared in beamer and they contain proofs of results which are particularly lengthy shorter proofs are contained in the notes. In mathematics, the notion of a set is a primitive notion. Compare this to the concept of modes of limited transposition found in messiaens theory. In section 2 youll remember that we looked at the treble and bass staves and the role of clefs. The formal axiomatic system of ordinary set theory zfc. Ordinal number the second number of the designmation of a prime form within the fortelist of set classes, representing the setclasses place in. Open access to the sep is made possible by a worldwide funding initiative. 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. Questions are bound to come up in any set theory course that cannot be answered mathematically, for example with a formal proof. For those of you new to abstract mathematics elementary does not mean simple though much of the material.
This is a continuation of the first lecture on sets. Basic set theory a set is a many that allows itself to be thought of as a one. For any sentential assignment fthere is a function fmapping the set. A to say that a is a member element of a, and we write a. A complement, or not a is the set of all elements not in a. The intersection of the set with members 1, 2, 3 together with the set with members 3, 4, 5 is the set that has only the 3. Because music employs a set of pitches ranging from low to high, the staff acts like a map for the notes allowing us to hear, read or write them as. As you go up a staff from note to note by step without skipping any notes going linespacelinespace, etc.
Computer science, being a science of the arti cial, has had many of its constructs and ideas inspired by set theory. Cantorian set theory is founded on the principles of extension and abstraction, described above. Motivation when you start reading these notes, the first thing you should be asking yourselves is what is set theory and why is it relevant. Software deployment software maintenance programming team opensource model. The notation and symbols for sets are based on the operations performed on them. A unified approach to algebraic set theory chapter 2. If you continue browsing the site, you agree to the use of cookies on this website. Indeed, the relationship between the people who study formal systems whether from the point of view of proof theory, model theory or category theory and the. But even more, set theory is the milieu in which mathematics takes place today.
The big questions cannot be dodged, and students will not brook a flippant or easy answer. The main text for the course will be the lecture notes, available at. Today this concept is being used in almost every branch of mathematics. Set operations and the laws of set theory the union of sets a and b is the set ab fx. Introduction to logic and set theory 202014 general course notes december 2, 20 these notes were prepared as an aid to the student. There is a section for theories, applications, and planning processes. The strong tradition, universality and neutrality of set theory make it rm common ground on which to provide uni cation between seemingly disparate areas and notations of computer science. The five horizontal lines on which the notes sit are called a staff. Basic concepts of set theory, functions and relations. Jee notes of set theory, an important chapter for jee mains and advanced. The intersection of sets a and b is the set a\b fx. They are not guaran teed to be comprehensive of the material covered in the course.
Write each note on the correct line using quarter notes. Set theory is indivisible from logic where computer science has its roots. Muted triangles are notated with the stopped symbol, or, on a singleline stave they may be notated with a crossshaped notehead. Notes introduction i weve already implicitly dealt with sets integers, z. Ling 310, adapted from umass ling 409, partee lecture notes march 1, 2006 p. We assume the reader is familiar with elementary set theory as it is used in mathematics today. Cantor initiated the study of set theory with his investigations on the cardinality of sets of real numbers.
Though propositional logic will prove a useful tool to describe certain aspects of meaning, like the reasoning in 1, it is a blunt. Typesetters introduction thesenotesprovideagreatintroductiontoaxiomaticsettheoryandtopicsthereinappropriate for a. Elements of a set can be just about anything from real physical objects to abstract mathematical objects. Sets fundamental to set theory is the notion of membership. A logic of meaning and synonymy, with fritz hamm, pdf file. An introduction to independence proofs by kenneth kunen, naive set theory by paul r. 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. Because music employs a set of pitches ranging from low to high, the staff acts like a map for the notesallowing us to hear, read or write them as.
Basically, the definition states it is a collection of elements. Elements of set theory by sidneyfelder in the words of georgcantor 18451918,the creator of set theory and the modern theory of the in. In this method a set is described by listing elements, separated by commas, within braces. Notes below b should have a stem on the right side facing up. Draw the treble clef at the beginning of the staff. To view the requested content, install the latest version of the ios software on your device. Pdf this document contains notes on set theory that i have used in some of my other documents and in some of my answers to questions on. Key theories, applications, and planning processes related to.
Sets are used to define the concepts of relations and functions. Complex issues arise in set theory more than any other area of pure mathematics. To transpose a set down by n half steps, subtract n from each pitch class in the set. The staff notes one of the most basic elements in music is the note. The encyclopedia now needs your support please read how you can help keep the encyclopedia free. A set is often described in the following two ways. If you get a number larger than 11 or smaller than 0, add or subtract 12 to get a valid pitch class number. Nonetheless, we shall now give a careful treatment of set theory if only to to allow the reader to become conversant with our notation. If a note can be drawn in more than one place on the staff, choose which one you want to draw.
Notes above b have stems facing down from the left side. Music notation developed from one single line, used in the early days of writing down church chants. The music staff or music stave as it is also called is the series of 5 lines and 4 spaces that music is written down on. Cynthia church produced the first electronic copy in december 2002. This alone assures the subject of a place prominent in human culture. Music notation and theory for intelligent beginners.
How to read music notes quick learn cheat sheets, page 5 steady beat an unchanging, continuous pulse rhythm a pattern of long and short notes and rests. Sets are fundamental discrete structures that form the basis of. Note that a function need not be classified in only one way, e. Lecture notes on probability theory and random processes. In these notes, we introduce examples of uncertainty and we explain how the theory models them. Set theory is a branch of mathematical logic that studies sets, which informally are collections of.
A rigorous analysis of set theory belongs to the foundations of mathematics and. Set theory is likely to be around long after most presentday programming languages have faded from memory. In maths, set theory was developed to explain about collections of objects. Introduction to set theory notes introduction to set theory, second edition revised and expanded, by karel hrbacuk and thomas jech, dekker 1984 copies of the classnotes are on the internet in pdf format as given below. Typically the students were advanced undergraduate mathematics majors, a few beginning graduate students in mathematics, and some graduate students from other areas that included. 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. For two sets, we may speak of whether or not one set is contained in the other. The collection of all states of the system form a space x, and the evolution is represented by either. If the indian cricket team wins a test match, then all. Rather, it is a menu of options for wic staff to consider when planning or implementing any training strategy.
A complement of a set is all of the elements in the universe that are not in the set. An important feature of a set is that its elements are \distinct or \uniquely identi able. 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. A subset of a set a is another set that contains only elements from the set a, but may not contain all the elements of a. Having considered rhythm, metre and rests in sections 3 and 4, we can now return to the discussion of pitch, begun in section 2, and examine how scales and key signatures are formed. Set theory and relations studypivot free download dpp. They can also be notated on a singleline staff with either a triangle shape or a normal notehead. James talmage adams produced a major revision in february 2005. Set theory is the axiomatization of mathematics, and without ac noone seriously doubts its truth, or at least consistency.
In particular, we will learn about the consequences of set operations and their business uses. If you believe that this message is in error, your browser may be running. Note name speed tests are race the clock music theory worksheets. If you were to let the set with 1,2,3 be set a, and the set with 3,4,5 be set b, then you could use venn diagrams to illustrate the situation see figure 2. The concept of set serves as a fundamental part of the present day mathematics. The complement of a set can be represented with several differing notations. These notes for a graduate course in set theory are on their way to becoming a book. 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. From sets and types to topology and analysistowards practicable.
Nb note bene it is almost never necessary in a mathematical proof to remember that a function is literally a set of ordered pairs. The first number in the designation of a prime form within the fortelist of setclasses. They are not guaranteed to be comprehensive of the material covered in the course. The set difference a minus b is the set of elements that are in a, with those that are in b subtracted out. Chapter 4 set theory \a set is a many that allows itself to be thought of as a one. If the interval vector of the set contains an entry equal to the number of notes in the set, or half that number in the case of the tritone, then the set will be transpositionally symmetrical. Basic set theory daniel murfet may 16, 2006 throughout we work with the foundation provided by standard zfc fct,section 3.
Other applications in analysis and geometry were found, and cantors theory of sets won. These notes were prepared using notes from the course taught by uri avraham, assaf hasson, and of course, matti rubin. In particular we do not assume we are working inside any. This short paper provides a summary of the tutorial on categorical logic given by the second named. This document does not judge which theory, application or process is the best or identify any one that should be used above the others. Set theory is also the most philosophical of all disciplines in mathematics. Georg cantor this chapter introduces set theory, mathematical induction, and formalizes the notion of mathematical functions. A unified approach to algebraic set theory homepages of uva.
1305 1405 1118 117 943 1416 836 888 1643 12 42 1606 941 866 113 830 621 1171 541 763 78 347 1618 377 1363 347 343 263 1308 18 1104 1451 1187 254 1121 1032 1479 1294 507 802 714 1078 43 556 1489 782 1365