These notes were prepared using notes from the course taught by uri avraham, assaf hasson, and of course, matti rubin. March 30, 2009 basic set theory we will use the standard notation for containments. Thus, the set a contains a finite number of elements. Basic set theory a set is a many that allows itself to be thought of as a one. 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. Pdf slides of lessons, 2629 october 2009 lemmensinstituut, leuven belgium. Basic concepts of set theory, functions and relations.
This section includes a full set of lecture notes for the course. The elements of a set may be manifested in music as simultaneous chords. Set theory in mathematics complete free digital video. This is the basic set theory that we follow in set theoretic. Pitch class sets, normal order, and transformations open. We have already seen in the notes on set classes that while in tonal theory. Get your kindle here, or download a free kindle reading app.
I offer no definition of what a set is beyond the intuitive notion described above. Introduction to set theory a solution manual forhrbacek and jech1999 jianfei shen school of economics, the university of new south wales sydney, australia. Clark ross consider and play the opening to schoenbergs three piano pieces, op. Set theory chiara ghidini fbkirst, trento, italy 20142015 chiara ghidini mathematical logic outline set theory relations functions 1 set theory basic concepts operations on sets operation properties 2 relations properties equivalence relation 3 functions properties chiara ghidini mathematical logic outline.
If there were a set asuch that x2 for all x, then is a set of all sets, which, as we have proven, does not exist. This note is an introduction to the zermelofraenkel set theory with choice zfc. Handbook of set theory university of california, irvine. We may also investigate combinatorial game theory, which is interested in games like chess or go. The fundamental concept of musical set theory is the musical set, which is an unordered collection of pitch classes rahn 1980, 27. More exactly, a pitchclass set is a numerical representation consisting of distinct integers i. Nb note bene it is almost never necessary in a mathematical proof to remember that a function is literally a set of ordered pairs. Complex issues arise in set theory more than any other area of pure mathematics.
After mathematical logic has provided us with the methods of reasoning we start with a very basic theory. Axiomatic set theory michael meyling july 30, 2011. Theres also an ant taskant tasks included that provides the librarys functionality. Although elementary set theory is wellknown and straightforward, the modern subject, axiomatic set theory, is both conceptually more di.
Basic set theory daniel murfet may 16, 2006 throughout we work with the foundation provided by standard zfc fct,section 3. Pdf set theory based analysis of atonal music researchgate. This book is dedicated to prabir roy, louis mcauley, jonathan. Wright school of mathematical sciences queen mary and west. Module 1 for teachers of primary and secondary mathematics. The purpose of this module is to introduce language for talking about sets, and some. Pitchclass set theory in music and mathematics by michael james mcneilis. There are other conceptions of set, but although they have genuine mathematical interest they are not our concern here. 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 is the axiomatization of mathematics, and without ac noone seriously doubts its truth, or at least consistency. The study of geometry, sequences, probability, etc. 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.
Informally, a class is a collection of objects, the involved objects are called the elements or members of the class. Sets, notational remarks, some axioms of zfc and their. Georg cantor this chapter introduces set theory, mathematical induction, and formalizes the notion of mathematical functions. For those of you new to abstract mathematics elementary does not mean simple though much of the material. 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. Circular permutation rotating the order of a pitch class set so that the second ordering permutation of the set begins with the second member. Set theory has experienced a rapid development in recent years, with major advances in forcing, inner models, large cardinals and descriptive set theory. Pdf from pitchclass set theory to musical composition. Class notes will generally be appended to the end of the pdf file after class. Free set theory books download ebooks online textbooks. Introduction to logic and set theory 202014 general course notes december 2, 20 these notes were prepared as an aid to the student. Introduction to modern set theory judith roitman december 6, 2011 1. For any sentential assignment fthere is a function fmapping the 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. The language of set theory and well formed formulas, classes vs. If we wish to understand how it is organized, we could begin by looking at the melody, which seems to naturally break. Ling 310, adapted from umass ling 409, partee lecture notes march 1, 2006 p. One of the basic properties of an ordinal is that it is a wellordered. This book is dedicated to prabir roy, louis mcauley, jonathan seldin, anil nerode, and maurice bo a, my teachers, and to w. Instead, i am going to show you what can be done with sets. Sets are used to define the concepts of relations and functions.
It is an extremely useful technique for composers to help understand and control the harmonies. Chapter 4 set theory nanyang technological university. Review of set theory michael williams last updated. These notations and methods can describe and manipulate any type of chord that can be created within a 12tone equally tempered scale. The general outcomes and standards for each unit as listed in alberta educations mathematics 302 assessment standards and exemplars document are also provided for your perusal. Introduction to set theory a solution manual forhrbacek and jech1999 jianfei shen school of economics, the university of new south wales. Pdf the soft set theory offers a general mathematical tool for dealing with uncertain, fuzzy, not clearly defined objects. The improving mathematics education in schools times project. If you are asked to notate specific pitches, you are free to choose any register.
The present book covers each of these areas, giving the reader an understanding of the ideas involved. 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. From pitch class set theory to musical composition. Set theory in mathematics complete free digital video tutorials.
Canadian mathematical bulletin kaplansky has a welldeserved reputation for his expository talents. Java library offering theory of setsset theory functionality complement, intersection, symetric difference, union, cartesian product and powerset. Set theory a mathematical model that we will use often is that of. Normal order is the most compressed way to write a given. Semantic scholar extracted view of portfolio of compositions. Colour television is based on the theory of additive colour mixing, where all colours including white can be created by mixing red, green, and blue lights read. An inventory of classes of musical objects, and of relations between these classes, its function was to prepare the ground for a model of largescale pitch organization in music of the twentieth century. Handbook of set theory foreman, kanamori, and magidor eds. In particular we do not assume we are working inside any.
In this page we discuss completely all the concepts of set theory that is required in school level. However far the axis is from 0, the inversionallypaired pitch class will always be the. Pdf it is a lecture note on a axiomatics set theory, zf set theory with ac, in short zfc. When we partition a set, we break it down into mutually exclusive and exhaustive regions, i. I make no guarantees whatsoever as to their completeness or correctness. Pitchclass set theory and its contexts eastman studies in music. The concept of set serves as a fundamental part of the present day mathematics. The venn diagram below should help you get the picture. Musical set theory provides concepts for categorizing musical objects and describing their.
As name implies it is the set with collection of all the elements and usually denoted by u. If you get a number larger than 11 or smaller than 0, add or subtract 12 to get a valid pitch class number. More exactly, a pitch class set is a numerical representation consisting of distinct integers i. Set 31 has three possible rotationsinversions, the normal form of which is the smallest pie or most compact form. Apr 22, 2019 in any application of set theory, all the sets under consideration will likely to be subsets of a fixed set called universal set.
Much of what we call pitch class set theory today originally served as theintroductionto a theory. To transpose a set up by n half steps, add n to each pitch class in the set. What appeals to me most about descriptive set theory is that to study it you must reallyunderstandso many things. They are not guaranteed to be comprehensive of the material covered in the course. Tonal, atonal and microtonal pitchclass categories springerlink. A set is an unordered collection of things not counting multiplicities, its elements. Any group of pitch classes can be a pitch class set. It is designed for a onesemester course in set theory at the advanced undergraduate or beginning.
When we talk about a group of pitch classes as a unit, we call that group a pitch class set, often abbreviated pc set. The theory of sets was developed by german mathematician georg cantor 18451918. Today this concept is being used in almost every branch of mathematics. Network theory complete notes ebook free download pdf. Pdf abstract the article presents basic posssibilities of interdisciplinary oriented research in analysis of atonal music through set theory. Introduction to logic and set theory202014 general course notes december 2, 20 these notes were prepared as an aid to the student. To transpose a set down by n half steps, subtract n from each pitch class in the set. Axiomatic set theory january 14, 20 1 introduction one of our main aims in this course is to prove the following. This book is dedicated to prabir roy, louis mcauley. It is a perfect classroom tool for core mathematics instruction and enrichment. Chapter 4 set theory \a set is a many that allows itself to be thought of as a one. Sets and elements set theory is a basis of modern mathematics, and notions of set theory are used in all formal descriptions. Handson math attribute blocks provides an interactive learning enviroment for learning and enhancing mathematical concepts related to set theory, geometry, number theory, and logical thiking. These books are made freely available by their respective authors and publishers.
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