58 CHAPTER 2 Sets To be eligible to vote in the United States, a person must be at least 18 years old and must be a U.S. citizen. We could define set A as the set of people who are at least 18 years old and set B as the set of people who are U.S. citizens. Then the set of eligible voters would be the set of people who belong to both set A and set B. Note that this is a very different set than the set of people who belong to set A or set B. In this section, we will use a tool that will allow us to better understand the difference between statements involving the word and and statements involving the word or. Venn Diagrams, Set Operations, and Data Representation SECTION 2.3 LEARNING GOALS Upon completion of this section, you will be able to: 7 Construct a Venn diagram with two sets. 7 Determine the complement of a set. 7 Determine the intersection of two sets. 7 Determine the union of two sets. 7 Understand the relationship between n A B n A nB ( ), ( ), (), < and n A B ( ). > 7 Determine the difference of two sets. 7 Determine the Cartesian product of two sets. Why This Is Important The words and and or play an important role in many everyday applications. In addition to determining voter eligibility, these applications may include the wording of employment offers, real estate contracts, and other legal documents. Knowing the proper interpretation of statements involving and and or may help you make wise legal and financial decisions. Construct Venn Diagrams with Two Sets A useful technique for organizing data into sets and illustrating set relationships is the Venn diagram, named for English mathematician John Venn (1834–1923). Venn invented the diagrams and used them to illustrate ideas in his text on symbolic logic, published in 1881. In a Venn diagram, a rectangle usually represents the universal set, U. The items inside the rectangle may be divided into subsets of the universal set. The subsets are usually represented by circles. In Fig. 2.1, the circle labeled A represents set A, which is a subset of the universal set. Two sets may be represented in a Venn diagram in any of four different ways, as shown in Fig. 2.2. Two sets A and B are disjoint when they have no elements in common. Two disjoint sets A and B are illustrated in Fig. 2.2(a). If set A is a proper subset of set B, A B, , the two sets may be illustrated as in Fig. 2.2(b). If set A contains U A Figure 2.1 Research Activity 65. Ladder of Life In this section, we discussed the ladder of life. Do research and indicate all the different classifications in the Linnaean system, from most general to the most specific, in which a koala belongs. 66. Data Mining In this section, we discussed data mining and set theory. Do research on how social media companies and other organizations use data mining. Write a report that includes several applications of data mining and how it affects your everyday life. *See Instructor Answer Appendix Vesperstock/Shutterstock
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