2.4 Venn Diagrams with Three Sets and Verification of Equality of Sets 71 You are looking for a career that best matches your strengths, which include adaptability, dependability, and honesty. Is there a way that you can determine which careers match all three of these strengths? How about only two of these strengths? The answer is yes! In this section, we will use Venn diagrams to answer similar questions. We will learn that a Venn diagram can be used to display important information quickly and efficiently. U A B III II I IV V VI VII VIII C Figure 2.15 SECTION 2.4 Venn Diagrams with Three Sets and Verification of Equality of Sets LEARNING GOALS Upon completion of this section, you will be able to: 7 Construct a Venn diagram with three sets. 7 Use Venn diagrams to verify equality of sets. 7 Understand De Morgan’s laws for sets. Why This Is Important Organizing sets of data using Venn diagrams often helps us to understand the relationship among various sets such as the sets of careers that require similar strengths. We will be happiest in a career that matches our strengths. For example, if one of your strengths is mathematics then you might want to work in science, engineering, or accounting. Using Venn diagrams is a valuable way to categorize your strengths and weaknesses when choosing a career or place to work. Construct a Venn Diagram with Three Sets In Section 2.3, we learned how to use Venn diagrams to illustrate two sets. Venn diagrams can also be used to illustrate three sets. For three sets, A B, , and C, the diagram is drawn so the three sets overlap (Fig. 2.15), creating eight regions. The diagrams in Fig. 2.16 emphasize selected regions of three intersecting sets. When constructing Venn diagrams with three sets, we generally start with region V and work outward, as explained in the following procedure. 101. A C> {2, 4, 6, 8, }, … or C 102. ′A C> { } In Exercises 111–114, determine the relationship between set A and set B if 111. = A B B. > B A# 112. = A B B. < A B# 113. = ∅ A B . > A and B are disjoint sets. 114. = ∅ A B . < Both A and B must be { }, and therefore they are equal sets. Research Activity 115. Fastest-Growing Occupations Use the U.S. Bureau of Labor Statistics website to create a Venn diagram illustrating the 10 fastest-growing occupations for college graduates with an Associate’s degree based on employment in 2021 and the estimated employment for those same occupations in 2031. 103. ′ B C C ( ) < < {0, 1, 2, 3, 4, }, … or U 104. ′ A C B ( ) > > {2, 6, 10, 14, 18, }… In Exercises 105–110, determine whether the answer is ∅, A, or U. (Assume ≠ ∅ ≠ A A U , .) 105. ′ A A > ∅ 106. ′ A A < U 107. ∅ A< A 108. ∅ A> ∅ 109. ′A U< U 110. A U> A Stockfour/Shutterstock
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