80 CHAPTER 2 Sets The members of a health club were surveyed about taking fitness classes at the club. Suppose the data obtained from the survey show how many members took a yoga class, how many members took a spinning class, and how many members took a class in yoga and a class in spinning. How can the manager of the club use this data to determine how many members took only a yoga class? In this section, we will learn how to use Venn diagrams to answer this type of question. Robert Kneschke/Shutterstock Set Applications and Survey Data Analysis SECTION 2.5 LEARNING GOAL Upon completion of this section, you will be able to: 7 Use Venn diagrams and set concepts to solve application problems. Why This Is Important As you read through this section, you will see many real-life applications of set theory. These applications can help us understand how to organize data into different sets. In addition to the health club application described above, we will also work with applications involving homeowners’ insurance policies, bookstore sales, restaurant surveys, and many other financial and legal applications. Knowing how to organize such data into sets is important for making decisions affecting businesses and our personal lives. We can solve practical problems involving sets by using the problem-solving process discussed in Chapter 1: Understand the problem, devise a plan, carry out the plan, and then examine and check the results. To help understand the problem, look for key phrases such as “only set A, ” “set A and set B, ” “set A or set B, ” “set A and set B and not set C.” Remember that and means intersection, or means union, and not means complement. In this section we will complete Venn diagrams by writing the number of elements in each region of the Venn diagram as opposed to writing the elements themselves, as we did in Section 2.4. For example, in Fig. 2.26, the number 80 is written in region II of the Venn diagram. This means that if we were to write out the elements that belonged to region II, as we did in Section 2.4, there would be 80 different elements written in region II. Also in this section, we will be solving problems that involve data obtained from surveys. Our plan to solve these problems, will be to first complete a Venn diagram and then answer the questions asked. When completing the Venn diagrams, we generally start with the intersection of the sets of data and then work outward. We demonstrate this process in Example 1. Example 1 Restaurant Customer Survey The restaurant Delizioso received the following data from 250 customer surveys. 175 purchased pizza. 141 purchased chicken wings. 80 purchased both pizza and chicken wings. Of those surveyed, how many customers a) did not purchase either pizza or chicken wings? b) purchased chicken wings but not pizza? c) purchased pizza but not chicken wings? d) purchased pizza or chicken wings? Timely Tip When drawing Venn diagrams, we generally start with the intersection of the sets and work outward.
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