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