52 CHAPTER 2 Sets Consider the following sets. Set A {baseball, basketball, = hockey}. Set B {baseball, football, basketball, hockey, = softball}. Note that each element of set A is also an element of set B. In this section, we will discuss how to illustrate the relationship between two sets, A and B, when each element of set A is also an element of set B. Subsets SECTION 2.2 LEARNING GOALS Upon completion of this section, you will be able to: 7 Determine and recognize subsets. 7 Determine and recognize proper subsets. 7 Determine the number of subsets and proper subsets of a given set. Why This Is Important The relationship between sets is important throughout life. For example, to gain a promotion at work, you may need to fulfill different sets of criteria or sets of goals. Organizing your goals into different sets may make it easier for you to meet these goals and fulfill the requirements for a promotion. Learning Catalytics Keyword: Angel-SOM-2.2 (See Preface for additional details.) MATHEMATICS TODAY Data Mining and Set Theory Data mining is the process of uncovering patterns and other valuable information from large data sets. One application of data mining is the gathering of massive amounts of data by social media companies from their users. The social media companies then either sell this data to other companies or use this data to uncover specific users’ interests and buying patterns. The information obtained is then used to target advertising at potential customers, maximizing the advertising effectiveness and generating sales. Data mining is also used by law enforcement agencies to detect fraud, identity theft, and other online criminal activity. Medical researchers use data mining to identify successful medical therapies for different illnesses. Researchers in many fields are continuing to discover more applications of data mining. Why This Is Important Data mining relies on the set theory concepts that we are studying in this chapter including sets, subsets, union, intersection, and complement. A basic understanding of set theory can help us understand the underlying strategies involved in data mining. Subsets In our complex world, we often break larger sets into smaller, more manageable sets, called subsets. Subsets play a large role when we use sets to help us organize data. For example, consider the set of people in your class. Suppose we categorize the set of people in your class according to the first letter of their last name (the A’s, B’s, C’s, etc.). When we do so, each of these sets may be considered a subset of the original set. Each of these subsets can be separated further. For example, the set of people whose last name begins with the letter A can be categorized as either male or female or by their age. Each of these collections of people is also a subset. A given set may have many different subsets. Definition: Subset Set A is a subset of set B, symbolized by A B, # if and only if all the elements of set A are also elements of set B. The symbol A B# indicates that “set A is a subset of set B.” The symbol Ü is used to indicate “is not a subset.” Thus, A BÜ indicates that set A is not a subset of set B. To show that set A is not a subset of set B, we must find at least one element of set A that is not an element of set B. Example 1 A Subset? Determine whether set A is a subset of set B. a) = A {Statue of Liberty, Gateway Arch, Space Needle} = B {Statue of Liberty, Gateway Arch, Golden Gate Bridge, Space Needle} b) = A {1, 3, 5, 7} = B {1, 3} c) = | A x x { is a yellow fruit} = | B x x { is a red fruit} d) = A {pink, purple, blue} = B {purple, blue, pink} Jim West/Alamy Stock Photo
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