48 CHAPTER 2 Sets Note that the empty set has cardinality of 0. Also note that { }∅ is not the empty set. This set contains the element ∅ and has a cardinality of 1. The set { }0 is also not the empty set because it contains the element 0. It also has a cardinality of 1. Example 12 Solution Set to an Equation Indicate the set of natural numbers that are solutions to the equation + = x 5 0. Solution The solutions to the equation are those numbers that make the equation a true statement. The only number that makes the equation a true statement is −5. Because −5 is not a natural number, the solution set is the empty set, { } or ∅. ■ Now try Exercise 35 Another important set is a universal set. Definition: Universal Set A universal set, symbolized by U, is a set that contains all the elements for any specific discussion. When a universal set is given, only the elements in the universal set may be considered when working the problem. If, for example, the universal set for a particular problem is defined as { } = U 1, 2, 3, 4, . . . , 10 , then only the natural numbers 1 through 10 may be used in that problem. Exercises Warm Up Exercises In Exercises 1–10, fill in the blank with an appropriate word, phrase, or symbol(s). 1. A collection of objects is called a(n) ________. Set 2. Three dots placed in a set to show that the set continues in the same manner is called a(n) ________. Ellipsis 3. The three ways a set can be written are ________, ________, and ________. Description, Roster form, Set-builder notation 4. A set is said to be ________ if it either contains no elements or the number of elements in the set is a natural number. Finite 5. A set that is not finite is called a(n) ________ set. Infinite 6. Two sets that contain the same number of elements are called ________ sets. Equivalent 7. Two sets that contain the same elements are called ________ sets. Equal 8. The number of elements in a set is called the ________ number of the set. Cardinal 9. The set that contains no elements is called the ________ set. Empty or null 10. The two ways to indicate an empty set are ________ and ________. { } ∅ , Practice the Skills In Exercises 11–16, determine whether each set is well defined or not well defined. 11. The set of best-tasting cheeses Not well defined 12. The set of funniest comedians Not well defined 13. The set of countries with a population of more than 1 trillion people Well defined 14. The set of Best Picture Academy Award winners Well defined SECTION 2.1 Instructor Resources for Section 2.1 in MyLab Math • Objective-Level Videos 2.1 • PowerPoint Lecture Slides 2.1 • MyLab Exercises and Assignments 2.1 KAKIMAGE2/Alamy Stock Photo
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