42 CHAPTER 2 Sets Throughout your day you may have many different tasks to complete. Going to class, going to work, making appointments, shopping, spending time with family and friends, responding to email, and checking Instagram are just some of the things you may do on a daily basis. In this section, we will discuss ways to sort or classify items, such as tasks we need to complete, into sets. We will also discuss different methods that can be used to indicate sets. SECTION 2.1 Set Concepts LEARNING GOALS Upon completion of this section, you will be able to: ■ Understand methods to indicate a set including description, roster form, and set-builder notation. ■ Understand fundamental set concepts including finite sets, infinite sets, equal sets, equivalent sets, the cardinal number of a set, the null or empty set, and universal sets. Profile in Mathematics Georg Cantor Georg Cantor (1845–1918), born in St. Petersburg, Russia, is recognized as the founder of set theory. Cantor’s creative work in mathematics was nearly lost when his father insisted that he become an engineer rather than a mathematician. His two major books on set theory, Foundations of General Theory of Aggregates and Contributions to the Founding of the Theory of Transfinite Numbers, were published in 1883 and 1895, respectively. More information on Cantor is available in Section 2.6, Infinite Sets. Why This Is Important Set classifications are important in a range of applications that affect our daily lives. Organizing data into sets can help us make better decisions. We encounter sets in many different ways every day of our lives. A set is a collection of objects, which are called elements or members of the set. For example, the United States is a collection, or set, of 50 states. The 50 individual states are the members or elements of the set that is called the United States. Sets are extremely useful tools when organizing information. Data refers to information that is collected and analyzed to help with decision making. In this chapter, we will use sets to help us organize data and to help understand how the data can help us make informed decisions. We will further analyze data in Chapter 12, Statistics . A set is well defined if its contents can be clearly determined. The set of U.S. presidents is a well-defined set because its contents, the presidents, can be named. The set of the three best movies is not a well-defined set because the word best is interpreted differently by different people. In this text, we use only well-defined sets. Example 1 Well-Defined Sets Determine whether the following sets are well-defined sets or not well-defined sets. a) The set of the best rock & roll bands of all time. b) The set of inductees into the Rock & Roll Hall of Fame. Solution a) Since the word best is open to interpretation by different people, the contents of this set cannot be clearly determined. Thus, this is not a well-defined set. b) Since the contents of this set can be clearly determined, this is a welldefined set. ■ Now try Exercise 11 Methods to Indicate a Set Three methods are commonly used to indicate a set: (1) description, (2) roster form, and (3) set-builder notation. The method of indicating a set by description is illustrated in Example 2. Milkos/123RF
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