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. The brief examples that follow illustrate how the definitions and preference. 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. Today this concept is being used in almost every branch of mathematics. In work on zf set theory, the notion of class is informal, whereas other set theories, such as nbg set. Pitchclass set theory and its contexts combines thorough discussions of musical concepts with an engaging historical narrative. Introduction to modern set theory judith roitman december 6, 2011 1. Here one regards mathematics as a formal game and one is only concerned with the question of consistency.
Theories of pitchclass set genera proceed from the general premise that within. A function takes input as elements of a set, performs some action on the set elements and gives output as another set. Here is the access download page of set theory problems and solutions pdf, click this link to download or read online. Sets are used to define the concepts of relations and functions. There are other conceptions of set, but although they have genuine mathematical interest they are not our concern here. It has been and is likely to continue to be a a source of fundamental ideas in computer science from theory to practice. Problems are sorted into different sets based on how hard they are to solve.
The study of geometry, sequences, probability, etc. Pitch class theory is treated here as part of the musical and cultural landscape of the united states. I offer no definition of what a set is beyond the intuitive notion described above. 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. Georg cantor this chapter introduces set theory, mathematical induction, and formalizes the notion of mathematical functions.
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. When we partition a set, we break it down into mutually exclusive and exhaustive regions, i. Equivalent pitchclass sets belong to the same set class. Complex issues arise in set theory more than any other area of pure mathematics. There is enough material in the text for a yearlong course at the upperundergraduate level. In set theory and its applications throughout mathematics, a class is a collection of sets or sometimes other mathematical objects which can be unambiguously defined by a property that all its members share. Introduction to set theory a solution manual forhrbacek and jech1999 jianfei shen school of economics, the university of new south wales sydney, australia. If you get a number larger than 11 or smaller than 0, add or subtract 12 to get a valid pitch class number. Im no expert in set theory, but here are some things that i know ill be working in zermelofraenkel set theory first off, a set is a class.
It enumerates the notes in the 12 et system and allows certain calculations to be carried out on sets of notes that would you not be able to easily able to do otherwise. The book will be a useful resource for both experts working in a relevant or adjacent area and beginners wanting to learn set theory via selfstudy. Chapter 4 set theory nanyang technological university. In work on zermelofraenkel set theory, the notion of class is informal, whereas other set theories, such. In mathematics, the notion of a set is a primitive notion. Chapter 4 set theory \a set is a many that allows itself to be thought of as a one. Pitch class sets an unordered collection of pitch classes that forms a basis for some kind of compositional activity used to generate harmonies or melodies. 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 has experienced a rapid development in recent years, with major advances in forcing, inner models, large cardinals and descriptive set theory. Set theory is the ultimate branch of mathematics ijoart. Cardinal number refers to the number of elements in a set. Cynthia church produced the first electronic copy in december 2002. The pedagogy of pitchclass set theory as a means of posttonal music. Basic concepts of set theory, functions and relations. To transpose a set up by n half steps, add n to each pitch class in the set. Get set theory problems and solutions pdf file for free from our online library pdf file. Pitch class a pitch without regard to its octave position. In this diagram, the set a the rectangle is partitioned into sets w,x, and y. For further examples of conway and hodgmans active learning techniques. If we generate a multiset, we will rewrite it as a set. In set theory, inversion means the same number of half steps in the opposite direction.
Set theory logic software hardware why do we have the same operator names in logic and software and. Learn about sets and the different functions of sets. Instructions specific to my java applet are written in green text. James talmage adams produced a major revision in february 2005. The use of economics in competition law 2005, jan 27, brussels the views expressed herein are not purported to reflect those of the federal trade commission, nor any of its commissioners. Introduction to logic and set theory202014 general course notes december 2, 20 these notes were prepared as an aid to the student. Musical set theory provides concepts for categorizing musical objects and describing their. The lord by wisdom founded the earth, by understanding he established the heavens. Set theory is indivisible from logic where computer science has its roots. Informally, a class is a collection of objects, the involved objects are called the elements or members of the class. Zeephongsekul a school of mathematics and geospatial sciences, rmit university, melbourne email. For those of you new to abstract mathematics elementary does not mean simple though much of the material. The venn diagram below should help you get the picture.
Ive dusted off the calculator code and developed a set theory app for ios. 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. To transpose a set down by n half steps, subtract n from each pitch class in 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. Computer science, being a science of the arti cial, has had many of its constructs and ideas inspired by set theory. 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.
They originated as handwritten notes in a course at the university of toronto given by prof. More exactly, a pitchclass set is a numerical representation consisting of distinct integers i. The most commonly referenced set classes are trichords. Usually it isa motive that occurs melodically, harmonically, or both. After brahms, tonality in western music began to break down. This document is intended as a primer for those who are new to musical set theory and as a companion to my settheory java applet. Handbook of set theory foreman, kanamori, and magidor eds. Our learning system helps your children learn at their own pace.
A concept like pitch, for example, is very concrete, while pitch class is somewhat more abstract. In work on zf set theory, the notion of class is informal, whereas other set theories, such as nbg set theory, axiomatize the notion of class. My children are unable to cope with the pace of topics being taught in class. What appeals to me most about descriptive set theory is that to study it you must reallyunderstandso many things. Set theory for computer science university of cambridge. Thus, the set a contains a finite number of elements. The elements of a set may be manifested in music as simultaneous chords, successive tones as in. Zrelation and homometry a pitch class set is a set of pitches where octaves are equivalent, and enharmon. It is not until he becomes aware of some of the difficulties in set theory that he would even begin to question it.
In any system, a class is a unary predicate a seen as the set of objects where a is true, that is the class of all x such that ax in a set theoretical universe, each set e is synonymous with the class of the x such that x. My child is interested in learning only some subjects. Instead, i am going to show you what can be done with sets. 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 class that is not a set is informally in zermelofraenkel called a proper class, and a class that is a set is sometimes called a small. The realist position is probably the one which most mathematicians would prefer to take. Set class and prime form 1 lots of concepts in pitch class set theory are best viewed along a sliding scale of concreteness or abstractness. It is designed for a onesemester course in set theory at the advanced undergraduate or beginning.
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. For further examples of conway and hodgmans active learning techniques, see. Although elementary set theory is wellknown and straightforward, the modern subject, axiomatic set theory, is both conceptually more di. From pitch class set theory to musical composition. Pitch class set theory is just an analytical tool that gives you a different way to look at sets of notes. More exactly, a pitch class set is a numerical representation consisting of distinct integers i. An acquaintance with various examples and properties of well.
A pitch class is the set of all pitches that can have the same name, for example, all the cs. In set theory, what is the difference between a set of. The classes that are not sets are called proper classes. The precise definition of class depends on foundational context. Pdf undergraduate music students often experience difficulties when. For shorter onesemester or onequarter courses, a variety of arrangements of topics are possible. The present book covers each of these areas, giving the reader an understanding of the ideas involved. Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects. Every set is a class, no matter which foundation is chosen. Sets and elements set theory is a basis of modern mathematics, and notions of set theory are used in all formal descriptions. A little set theory never hurt anybody matthew saltzman department of mathematical sciences clemson university draft. They are not guaranteed to be comprehensive of the material covered in the course.
For any sentential assignment fthere is a function fmapping the set. Axiomatic set theory january 14, 20 1 introduction one of our main aims in this course is to prove the following. Introduction to logic and set theory 202014 general course notes december 2, 20 these notes were prepared as an aid to the student. The concept of set serves as a fundamental part of the present day mathematics. Ling 310, adapted from umass ling 409, partee lecture notes march 1, 2006 p. A collection of well defined and distinct objects is called a set. No matter what orderform they take in the music, they will retain their pitch class and interval class identity. They can repeat the lessons as many times as required. Dec 21, 2011 im no expert in set theory, but here are some things that i know ill be working in zermelofraenkel set theory first off, a set is a class. B for example, if s is the set of all juices in the supermarket, and t is the set of all. The theory of sets was developed by german mathematician georg cantor 18451918.
These notes for a graduate course in set theory are on their way to becoming a book. Pdf from pitch class set theory to musical composition. The fundamental concept of musical set theory is the musical set, which is an unordered collection of pitch classes rahn 1980, 27. We can perform a pitch, but we cant really perform a pitch class. Set theory is the ultimate branch of mathematics mohammad al amin lecture mathematics department of business administration z. Central to the development of the type concept has been its close relationship with set theory to begin with and later its even more intimate relationship with category theory. These notes were prepared using notes from the course taught by uri avraham, assaf hasson, and of course, matti rubin. Pitchclass sets are the basic building blocks of much posttonal music. The spine of the settheoretic universe, and the most essential class of objects in the study of set theory, is the class of ordinals. In set theory and its applications throughout mathematics, a class is a collection of sets or sometimes other mathematical objects that can be unambiguously defined by a property that all its members share. Bell this essay is an attempt to sketch the evolution of type theory from its beginnings early in the last century to the present day. Citations 0 references 0 this research hasnt been cited in any other publications. One of the basic properties of an ordinal is that it is a wellordered set. Learnhive icse grade 8 mathematics set theory lessons.
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