We would like our length scale to be smaller than the regional variation in the tissue, fig. The purpose of this module is to introduce language for talking about sets, and some. An introduction to set theory and topology open scholarship. 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.
The concept of sets is an essential foundation for various other topics in mathematics. 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 set of reals, \\mathbbr\, is the universal set in real analysis, and the set of complex numbers. A few elementary examples are the set of natural numbers. 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. Modern mathematics starts with the basic and undefinable concept of set. 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 abounds in problems that yet simple to state, are very hard to solve. A little set theory never hurt anybody matthew saltzman department of mathematical sciences clemson university draft. A standard notation is to specify a vector simply by the coordinates of the point it locates. They are called whole numbers or nonnegative integers. Sets may be thought of as a mathematical way to represent collections or groups of objects.
As the range of applications expanded, for example to the theory of plates and shells, some physical approaches failed and some succeeded. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics. Basic set theory is generally given a brief overview in courses on analysis, algebra, or topology, even though it is sufficiently important. A set is a many that allows itself to be thought of as a one. A subset is a set contained within another set, or it can be the entire set itself. Brouwer was an invited speaker of the icm in 1908 at rome and in 1912 at cambridge, uk. Introduction and preliminaries mathematics libretexts. 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.
The notion of set is taken as undefined, primitive, or basic, so. For us, a set will simply be an unordered collection of objects. Set theory, however, was founded by a single paper in 1874 by georg cantor. Basic concepts of set theory, functions and relations 0. Axioms and set theory mathematics university of waterloo.
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. Pure and algorithmic graph theory are the topics of bm76 and gib85, respectively. Some numbertheoretic problems that are yet unsolved are. Traditional courses on applied mathematics have emphasized problem solving techniques rather than the systematic development of concepts. It has been and is likely to continue to be a a source of fundamental ideas in computer science from theory to practice. Preliminaries from partee 1979, fundamentals of mathematics for linguistics xeroxed. Uwmilwaukee department of mathematical sciences 69,194 views. 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. Basic set theory a set is a many that allows itself to be thought of as a one. Complexity what is the root of the hardness can a less than perfect solution suffice. Set theory previous signed numbers fractions and percents. In any application of set theory there is an underlying universal set whose subsets are the objects of study. Generally, vectors are designated by a boldface symbol, e. Then we have i2i a i c \ i2i ac i and \ i2i a i c i2i ac i.
Mathematical preliminaries part 1 welcome to cs 766qic 820 theory of quantum information. See xeroxed section preliminaries from partee 1979. Vowels in the english alphabet v a, e, i, o, u first seven prime numbers. 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. 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. Personal taste has led me to include a few special but still wellknown topics. We will learn the basic properties of sets, how to define the size of a set, and how to compare. Set theory is a basis of modern mathematics, and notions of set theory are used in all formal descriptions. As a result, it is possible for such courses to become terminal mathematics courses rather than.
We think of a set as a structureless family, or collection, of objects. Also preliminaries from partee 1979, fundamentals of mathematics for linguistics. The set of all inputs for a function is called the domain. Set theory is seen as the foundation from which virtually all of mathematics can be derived.
In this chapter we will outline the background needed for a course in abstract algebra. The following lectures employ a number of concepts in topology, real analysis, and modern algebra. Basic concepts of set theory, functions and relations. When the subset is missing some elements that are in the set it is being compared to, it is a proper subset. Set which assigns to any topological space x its underlying set. Subsets a set a is a subset of a set b iff every element of a is also an element of b. 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. Sets and elements set theory is a basis of modern mathematics, and notions of. Preliminaries from partee 1979, fundamentals of mathematics for. One of the main applications of naive set theory is constructing relations. Set theory is a basis of modern mathematics, and notions of set theory are used. Functions a function is a rule that assigns each input exactly one output. Chapter 1 conceptual and mathematical preliminaries.
Mathematical topics typically emerge and evolve through interactions among many researchers. 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. For those of you new to abstract mathematics elementary does not mean simple though much of the material. X 2, 3, 5, 7, 11, 17 cs 441 discrete mathematics for cs m. 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 size of a set s, denoted jsj, is known as its cardinality or order. An introduction to set theory department of mathematics. Preliminaries a certain amount of mathematical maturity is necessary to find and study applications of abstract algebra. Set theory is a basis of modern mathematics, and notions of set theory are. Basic concepts of set theory, functions and relations 1. An introduction to set theory university of toronto.
Examples of alphabets include the binary alphabet f 0. Computer science, being a science of the arti cial, has had many of its constructs and ideas inspired by set theory. Rather than continuing to specify vectors in this clumsy way, we introduce a notation for them. In the first we give a description of the basic theorems of elementary catastrophe theory, along with heuristic explanations of why these theorems are valid. The language of set theory can be used to define nearly all mathematical objects. Basic mathematical notation and argument, including proof by contradiction, mathematical induction and its variants. Set theory is indivisible from logic where computer science has its roots. The rst section focuses on linear algebra, and the second on analysis and related topics. Mathematical preliminaries this course is about the fundamental capabilities and limitations of computers. 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. There are occasional hints at the rst and the fourth, leaving the others largely untouched. Mathematical preliminaries this book views nonlinear dynamical systems as mathematical systems. Each thing in the collection is called an element or member the set. Let a i i2i be a nonempty indexed collection of subsets a i.
When the subset is the set itself, it is an improper subset. Introduction to modern set theory judith roitman december 6, 2011 1. This chapter introduces set theory, mathematical in. When we describe a set by listing its elements, we enclose the list in curly braces. Preliminaries and basic operations properties of basic mathematical operations. Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects. Sets and elements set theory is a basis of modern mathematics, and notions of set theory are used in all formal descriptions. It is designed for a onesemester course in set theory at the advanced undergraduate or beginning graduate level. 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. Jun 23, 2019 for us, a set will simply be an unordered collection of objects. But even more, set theory is the milieu in which mathematics takes place today. Number theory is one of the oldest and most beautiful branches of mathematics.
This course is about the fundamental capabilities and limitations of. Textbook examples will serve as solution models to most of the exercise questions at the end. Set theory some basics and a glimpse of some advanced techniques. Basic set theory mathematical association of america. As the range of applications expanded, for example to the theory of plates and shells, some physical approaches failed and. Even more important is the ability to read and understand mathematical proofs. The first chapter in this part is a very rapid about 15 pages long overview of point set topology, essentially devoid of proofs except for one or two results where some brief hint of a proof is. He was the founder of the mathematical philosophy of intuitionism.
This universal set is usually clear from the context. The concept of a set is one of the most fundamental in mathematics. Set theory preliminaries the following basic results of set theory tell how unions and intersections behave under complements. A little set theory never hurt anybody mathematical sciences. 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.
The members of a set are referred to as its elements. Set theory for computer science university of cambridge. The order of the elements in a set doesnt contribute. Fundamentals of mathematics i department of mathematical. Its probably easier to get a feel for ordinals by looking at some examples of ordinals. A set is a group of objects, numbers, and so forth. The set of all allowable outputs is called the codomain. A basic knowledge of set theory, mathematical induction, equivalence relations, and matrices is a must.
A set is collection of distinct elements, where the order in which the elements are listed does not matter. For a more complete treatment of propositional and firstorder logics, we recommend fit90, gal86, and men79. Georg cantor this chapter introduces set theory, mathematical induction, and formalizes the notion of mathematical functions. This alone assures the subject of a place prominent in human culture. Brouwer 18811966 was a dutch mathematician and philosopher, who worked in topology, set theory, measure theory, and complex analysis. For example, in the study of the properties of integers, the set of integers, denoted by \\mathbbz\, is the universal set. Today we introduce set theory, elements, and how to build sets.
Duplicates dont contribute anythi ng new to a set, so remove them. The last three examples suggest that a set with elements has 2 subsets why. Cs 766qic 820 theory of quantum information fall 2011 lecture 1. In mathematics, a set is a welldefined collection of distinct objects, considered as an object in its own right. In this project we will learn elementary set theory from the original historical sources by two.
If you require more indepth explanations of these concepts, you can visit the wolfram math. Automata models of computation these are data as well as programs 2. Preliminaries and basic mathematical tools sciencedirect. Georg cantor this chapter introduces set theory, mathematical in duction, and formalizes the notion of mathematical functions. Goldbachs conjecture is every even integer greater than 2 the sum of distinct primes. 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. 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 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. 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. Halmos treatment of introductory set theory hal60 is both elegant and accessible.
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