44 CHAPTER 2 Sets The symbol ,∈ read “is an element of,” is used to indicate membership in a set. In Example 4, because 8 is an element of set A, we write ∈A 8 . This may also be written { } ∈8 6, 7, 8 . We may also write A 5 , ∉ meaning that 5 is not an element of set A. Set-builder notation (sometimes called set-generator notation) may be used to symbolize a set. Set-builder notation is frequently used in algebra. The following example illustrates its form and how it is read. = | D x { Condition(s) } Set D is the set of all elements x such that the condition(s) x must meet in order to be a member of the set. Consider E x x N x and 10 . { } = | ∈ > The statement is read: “Set E is the set of all the elements x such that x is a natural number and x is greater than 10.” The conditions that x must meet to be a member of the set are ∈ x N, which means that x must be a natural number, and > x 10, which means that x must be greater than 10. The numbers that meet both conditions are the set of natural numbers greater than 10. Set E in roster form is { } = … E 11, 12, 13, 14, Example 5 Using Set-Builder Notation a) Write set { } = B 1, 2, 3, 4, 5 in set-builder notation. b) Write, in words, how you would read set B in set-builder notation. Solution a) Because set B consists of the natural numbers less than 6, we write = | ∈ < B x x N x { and 6}. Another acceptable answer is = | ∈ ≤ B x x N x { and 5}. b) Set B is the set of all elements x such that x is a natural number and x is less than 6. Another acceptable answer is set B is the set of all elements x such that x is a natural number and x is less than or equal to 5. ■ Now try Exercise 39 Learning Catalytics Keyword: Angel-SOM-2.1 (See Preface for additional details.) Example 6 Set-Builder Notation to Roster Form Write set = | ∈ ≤ < A x x N x { and 2 8} in roster form. Solution { } = A 2, 3, 4, 5, 6, 7 ■ Now try Exercise 31 Example 7 Elements of a Set Determine whether each statement is true or false. If false, give the reason. a) { } ∈ blue orange, yellow, blue, green b) { } ∈ peppermint spearmint, cinnamon, licorice c) { } { } ∈ sapphire gold, diamond, ruby, sapphire
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