Sets-F09 - Chapter 1 Sets C HAPTER 1 SETS SETS I....

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Chapter 1 Sets 1 HAPTER 1 I. DEFINITION OF A SET We begin our study of probability with the discussion of the basic concept of set . We assume that there is a common understanding of what is meant by the notion of “a collection of things”, “a set of objects”, or “a group of articles”. Therefore we will not attempt to define this fundamental idea of a “collection of things”, but we will use it to give a definition of the mathematical usage of the term set . What is a set ? A set is a well-defined collection of distinct objects. The objects in the set are called the elements or members of the set. Capital letters A,B,C,… are usually used to denote sets and lowercase letters a,b,c ,… to denote the elements of a set. As we shall see by some examples, it is not enough to say that a set is simply a collection of objects. The distinguishing idea is that the collection must be well- defined . That is, for a collection S to be set it must always be possible to determine whether or not some given object x is an element of S . Examples 1.1: 1. The collection of the vowels in the word “probability”. 2. The collection of real numbers that satisfy the equation 0 9 2 = - x . 3. The collection of two-digit positive integers that are divisible by 5. 4. The collection of great football players in the National Football League. 5. The collection of intelligent members of the United States Congress. The first three collections are sets because in each case it is certainly possible to determine whether or not some given object x is an element of the set. On the other hand, collections 4 and 5 are not sets; for a given NFL football player x some people might consider him to be great, others might not. Similarly, different people would probably come up with different collections of “intelligent” members of Congress. Collection 5 also suggests the need for a special set, namely the set that has no elements. The set with no elements is called the empty set or the null set . The empty C SETS SETS SETS

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Chapter 1 Sets 2 set is denoted by the symbol Ø . For a concrete example, let B be the set of real numbers x that satisfy the equation 0 1 2 = + x , which is the same as 1 2 - = x . Since the square of any real number is always a nonnegative real number, there are no real numbers which satisfy this equation. Thus B is the empty set ( B is Ø ). Other examples which justify the need for the empty set will be given below. Specifying a set. There are various ways to denote a set. When the number of the elements in the set is small, or when the elements have an obvious pattern, you can simply list the elements, enclosed within braces. This method is called the roster method . Examples 1.2:
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This note was uploaded on 01/06/2011 for the course MATH 3333 taught by Professor Staff during the Spring '08 term at University of Houston.

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Sets-F09 - Chapter 1 Sets C HAPTER 1 SETS SETS I....

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