File Name: set language and notation .zip
Every field of study seeks a common terminology and symbology.
Basic set notation
Every field of study seeks a common terminology and symbology. While it is possible to think about a subject without knowing its particular language, it is not possible to communicate with others about that subject without some common frame of reference. Thus we begin with the basic terms and notations of set theory.
We begin this chapter with a brief description of discrete mathematics. We then cover some of the basic set language and notation that will be used throughout the text. Venn diagrams will be introduced in order to give the reader a clear picture of set operations. In addition, we will describe the binary representation of positive integers and introduce summation notation and its generalizations.
The term set is intuitively understood by most people to mean a collection of objects that are called elements of the set. This concept is the starting point on which we will build more complex ideas, much as in geometry where the concepts of point and line are left undefined.
Because a set is such a simple notion, you may be surprised to learn that it is one of the most difficult concepts for mathematicians to define to their own liking. For example, the description above is not a proper definition because it requires the definition of a collection. How would you define "collection"? Even deeper problems arise when you consider the possibility that a set could contain itself.
Although these problems are of real concern to some mathematicians, they will not be of any concern to us. Our first concern will be how to describe a set; that is, how do we most conveniently describe a set and the elements that are in it?
If we are going to discuss a set for any length of time, we usually give it a name in the form of a capital letter or occasionally some other symbol. In discussing set , if is an element of , then we will write. The most convenient way of describing the elements of a set will vary depending on the specific set.
When the elements of a set are enumerated or listed it is traditional to enclose them in braces. The choice of a set name is much like the choice of an identifier name in programming. Some large sets can be enumerated without actually listing all the elements. The three consecutive "dots" are called an ellipsis. We use them when it is clear what elements are included but not listed.
An ellipsis is used in two other situations. If we want to list a more general set such as the integers between 1 and , where is some undetermined positive integer, we might write. Standard Symbols. Sets that are frequently encountered are usually given symbols that are reserved for them alone.
A few of the other sets of numbers that we will use frequently are:. Set-Builder Notation. Another way of describing sets is to use set-builder notation. For example, we could define the rational numbers as. The important fact to keep in mind in set notation, or in any mathematical notation, is that it is meant to be a help, not a hindrance. We hope that notation will assist us in a more complete understanding of the collection of objects under consideration and will enable us to describe it in a concise manner.
However, brevity of notation is not the aim of sets. Also, there are frequently many different, and equally good, ways of describing sets. For example,. A proper definition of the real numbers is beyond the scope of this text. It is sufficient to think of the real numbers as the set of points on a number line.
The complex numbers can be defined using set-builder notation as , where. In the following definition, we will leave the word "finite" undefined. Definition 1. Any set that is not finite is an infinite set. Let be a finite set. The cardinality of a finite set A is denoted. As we will see later, there are different infinite cardinalities. We can't make this distinction until Chapter 7, so we will restrict cardinality to finite sets for now. We say that is a subset of if and only if every element of is an element of.
Example 1. In Example 1. Note that the ordering of the elements is unimportant. The number of times that an element appears in an enumeration doesn't affect a set. For example, if and , then. Warning to readers of other texts: Some books introduce the concept of a multiset, in which the number of occurrences of an element matters. A few comments are in order about the expression "if and only if" as used in our definitions. This expression means "is equivalent to saying", or more exactly, that the word or concept being defined can at any time be replaced by the defining expression.
Conversely, the expression that defines the word or concept can be replaced by the word. Occasionally there is a need to discuss the set that contains no elements, namely the empty set, which is denoted by.
This set is also called the null set. It is clear, we hope, from the definition of a subset, that given any set we have and. If is nonempty, then is called an improper subset of. All other subsets of , including the empty set, are called proper subsets of. The empty set is an improper subset of itself. Note 1. In fact earlier editions of this book sided with those who considered the empty set an improper subset. However, we bow to the emerging consensus at this time.
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Set Notation and Relations Every field of study seeks a common terminology and symbology. Chapter 1: Set Theory empty set Betty's math teacher said, in a sweat: "I will teach you some set theory yet! A true empty set! Jump to
01 Set language and notation.pdf
All of mathematics can be seen as the study of relations between collections of objects by rigorous rational arguments. More often than not the patterns in those collections and their relations are more important than the nature of the objects themselves. In mathematics, the collections are usually called sets and the objects are called the elements of the set. Functions are the most common type of relation between sets and their elements and the primary objects of study in Analysis are functions having to do with the set of real numbers. A set is an unordered collection of distinct objects, which we call its elements. Here are a few examples.
First we specify a common property among "things" we define this word later and then we gather up all the "things" that have this common property. There is a fairly simple notation for sets. We simply list each element or "member" separated by a comma, and then put some curly brackets around the whole thing:. The three dots OK, there isn't really an infinite amount of things you could wear, but I'm not entirely sure about that!
Introduction to Sets
It is natural for us to classify items into groups, or sets, and consider how those sets overlap with each other. We can use these sets understand relationships between groups, and to analyze survey data. An art collector might own a collection of paintings, while a music lover might keep a collection of CDs. Any collection of items can form a set.
When all sets under consideration are considered to be subsets of a given set U , the absolute complement of A is the set of elements in U , but not in A. If A is a set, then the absolute complement of A or simply the complement of A is the set of elements not in A within a larger set that is implicitly defined. In other words, let U be a set that contains all the elements under study; if there is no need to mention U , either because it has been previously specified, or it is obvious and unique, then the absolute complement of A is the relative complement of A in U : . The absolute complement of A is usually denoted by A c. Let A and B be two sets in a universe U.