On a property of the collection of all real algebraic numbers since the 5th century bc, beginning with greek mathematician zeno of elea in the west and early indian mathematicians in the east, mathematicians had struggled. Mathematical preliminaries part 1 welcome to cs 766qic 820 theory of quantum information. The goal of this lecture, as well as the next, is to present a brief overview of some of the basic mathematical concepts and tools that will be important in subsequent lectures of the. A set is a group of objects, numbers, and so forth.
We would like our length scale to be smaller than the regional variation in the tissue, fig. In this chapter we will outline the background needed for a course in abstract algebra. Set theory is a basis of modern mathematics, and notions of set theory are used in all formal descriptions. For example, in the study of the properties of integers, the set of integers, denoted by \\mathbbz\, is the universal set. 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. Set theory previous signed numbers fractions and percents. This series of lessons cover the essential concepts of math set theory the basic ways of describing sets, use of set notation, finite sets, infinite sets, empty. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics. Automata models of computation these are data as well as programs 2. For example, the numbers 2, 4, and 6 are distinct objects when considered separately, but when they are considered collectively they form a single set of size three, written 2, 4, 6. In this project we will learn elementary set theory from the original historical sources by two. A set is collection of distinct elements, where the order in which the elements are listed does not matter. It assumes no knowledge of logic, and no knowledge of set theory beyond the vague familiarity with curly brackets, union and intersection usually expected of an advanced mathematics. It abounds in problems that yet simple to state, are very hard to solve.
The concept of sets is an essential foundation for various other topics in mathematics. We think of a set as a structureless family, or collection, of objects. This video is an updated version of the original video released over two years ago. These objects are sometimes called elements or members of the set. Set theory is a basis of modern mathematics, and notions of set theory are used. Introduction to modern set theory judith roitman december 6, 2011 1. Basic concepts of set theory, functions and relations. The members of a set are referred to as its elements. Cs 766qic 820 theory of quantum information fall 2011 lecture 1. For a more complete treatment of propositional and firstorder logics, we recommend fit90, gal86, and men79.
The main notions of set theory cardinals, ordinals, transfinite induction are fundamental to all mathematicians, not only to those who specialize in mathematical logic or settheoretic topology. Mathematical topics typically emerge and evolve through interactions among many researchers. Its probably easier to get a feel for ordinals by looking at some examples of ordinals. Even more important is the ability to read and understand mathematical proofs. An introduction to set theory university of toronto. Goldbachs conjecture is every even integer greater than 2 the sum of distinct primes. This alone assures the subject of a place prominent in human culture. 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. The last three examples suggest that a set with elements has 2 subsets why.
As the range of applications expanded, for example to the theory of plates and shells, some physical approaches failed and some succeeded. Examples of alphabets include the binary alphabet f 0. Today we introduce set theory, elements, and how to build sets. Part ii of the text covers some advanced topics in set theory and also looks at ways in which set theory is applied to other areas of mathematics. One of the main applications of naive set theory is constructing relations. They are called whole numbers or nonnegative integers. The set of all inputs for a function is called the domain. For us, a set will simply be an unordered collection of objects. Georg cantor this chapter introduces set theory, mathematical induction, and formalizes the notion of mathematical functions. A set is a many that allows itself to be thought of as a one. A little set theory never hurt anybody mathematical sciences. The language of set theory can be used to define nearly all mathematical objects. Two sets are equal if and only if they have the same elements.
Axioms and set theory mathematics university of waterloo. Generally, vectors are designated by a boldface symbol, e. As the range of applications expanded, for example to the theory of plates and shells, some physical approaches failed and. Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects. There are occasional hints at the rst and the fourth, leaving the others largely untouched. This chapter introduces set theory, mathematical in. The following lectures employ a number of concepts in topology, real analysis, and modern algebra.
A subset is a set contained within another set, or it can be the entire set itself. Set theory is indivisible from logic where computer science has its roots. The first covers pure set theory, including the basic notions, order and wellfoundedness, cardinal numbers, the ordinals, and the axiom of choice and some of its consequences. If you require more indepth explanations of these concepts, you can visit the wolfram math. Basic concepts of set theory, functions and relations 1. Textbook examples will serve as solution models to most of the exercise questions at the end. For those of you new to abstract mathematics elementary does not mean simple though much of the material. When set theory is applied to the rest of mathematics, the methodology often used is to reduce the original question to a question in the area known as in nite combinatorics.
The notion of set is taken as undefined, primitive, or basic, so. Sets and elements set theory is a basis of modern mathematics, and notions of set theory are used in all formal descriptions. Set theory is seen as the foundation from which virtually all of mathematics can be derived. This course is about the fundamental capabilities and limitations of. But even more, set theory is the milieu in which mathematics takes place today. Basic set theory is generally given a brief overview in courses on analysis, algebra, or topology, even though it is sufficiently important, interesting, and simple to merit its own leisurely treatment. Computer science, being a science of the arti cial, has had many of its constructs and ideas inspired by set theory. The purpose of this module is to introduce language for talking about sets, and some. Traditional courses on applied mathematics have emphasized problem solving techniques rather than the systematic development of concepts. Order theory mark dean lecture notes for fall 2015 phd class in decision theory brown university 1 introduction we are going to begin the class with some mathematical preliminaries. Examples of this include sections on the statistical mechanical theory of polymer chains and the lattice theory of crystalline solids in the discussion of constitutive. The set of all allowable outputs is called the codomain. For example, structures in abstract algebra, such as groups, fields and rings, are sets closed under one or more operations. X 2, 3, 5, 7, 11, 17 cs 441 discrete mathematics for cs m.
Brouwer 18811966 was a dutch mathematician and philosopher, who worked in topology, set theory, measure theory, and complex analysis. We will learn the basic properties of sets, how to define the size of a set, and how to compare. Let a i i2i be a nonempty indexed collection of subsets a i. Basic mathematical notation and argument, including proof by contradiction, mathematical induction and its variants. Preliminaries and basic mathematical tools sciencedirect. Set theory some basics and a glimpse of some advanced techniques. Then we have i2i a i c \ i2i ac i and \ i2i a i c i2i ac i. Preliminaries from partee 1979, fundamentals of mathematics for. A standard notation is to specify a vector simply by the coordinates of the point it locates. Complexity what is the root of the hardness can a less than perfect solution suffice. He was the founder of the mathematical philosophy of intuitionism. 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.
It has been and is likely to continue to be a a source of fundamental ideas in computer science from theory to practice. Basic set theory a set is a many that allows itself to be thought of as a one. Preliminaries and basic operations properties of basic mathematical operations. The size of a set s, denoted jsj, is known as its cardinality or order. Jun 23, 2019 for us, a set will simply be an unordered collection of objects. The concept of a set is one of the most fundamental in mathematics. Also preliminaries from partee 1979, fundamentals of mathematics for linguistics.
The rst section focuses on linear algebra, and the second on analysis and related topics. Chapter 1 conceptual and mathematical preliminaries. Two of the basic techniques of set theory are trans nite induction and trans nite recursion, which are grounded in induction and recursion on the natural nubmers. When the subset is the set itself, it is an improper subset. Preliminaries from partee 1979, fundamentals of mathematics for linguistics xeroxed. The order of the elements in a set doesnt contribute.
When we describe a set by listing its elements, we enclose the list in curly braces. In any application of set theory there is an underlying universal set whose subsets are the objects of study. For example, a deck of cards, every student enrolled in math 103, the collection of all even integers, these are all examples of sets of things. An introduction to set theory department of mathematics. Georg cantor this chapter introduces set theory, mathematical in duction, and formalizes the notion of mathematical functions. Set theory, however, was founded by a single paper in 1874 by georg cantor.
Basic set theory mathematical association of america. Each thing in the collection is called an element or member the set. When the subset is missing some elements that are in the set it is being compared to, it is a proper subset. Subsets a set a is a subset of a set b iff every element of a is also an element of b. Brouwer was an invited speaker of the icm in 1908 at rome and in 1912 at cambridge, uk. Some numbertheoretic problems that are yet unsolved are. Rather than continuing to specify vectors in this clumsy way, we introduce a notation for them. Set theory is a basis of modern mathematics, and notions of set theory are. Basic concepts of set theory, functions and relations 0. In mathematics, a set is a welldefined collection of distinct objects, considered as an object in its own right. See xeroxed section preliminaries from partee 1979. Mathematical preliminaries this course is about the fundamental capabilities and limitations of computers. Modern mathematics starts with the basic and undefinable concept of set. Sets and elements set theory is a basis of modern mathematics, and notions of.
Preliminaries a certain amount of mathematical maturity is necessary to find and study applications of abstract algebra. A little set theory never hurt anybody matthew saltzman department of mathematical sciences clemson university draft. We speak, for example, of the set of students in a college, of men in a city, of women working for a corporation, of vectors in space, of points in a plane, or of events in the continuum of spacetime. The set 1,2 is a subset of the set 1,2,3, and the set 1,2,3 is a subset of the set 1,2,3. A few elementary examples are the set of natural numbers. Set theory preliminaries the following basic results of set theory tell how unions and intersections behave under complements.
It is designed for a onesemester course in set theory at the advanced undergraduate or beginning graduate level. As a result, it is possible for such courses to become terminal mathematics courses rather than. Introduction and preliminaries mathematics libretexts. Personal taste has led me to include a few special but still wellknown topics. Number theory is one of the oldest and most beautiful branches of mathematics. Basic set theory is generally given a brief overview in courses on analysis, algebra, or topology, even though it is sufficiently important. A basic knowledge of set theory, mathematical induction, equivalence relations, and matrices is a must. Pure and algorithmic graph theory are the topics of bm76 and gib85, respectively. Mathematical preliminaries this book views nonlinear dynamical systems as mathematical systems. Set theory for computer science university of cambridge. Vowels in the english alphabet v a, e, i, o, u first seven prime numbers. Uwmilwaukee department of mathematical sciences 69,194 views. Halmos treatment of introductory set theory hal60 is both elegant and accessible.
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