1 Work with Sets A set is a well-defined collection of distinct objects. The objects of a set are called its elements . By well-defined , we mean that there is a rule that enables us to determine whether a given object is an element of the set. If a set has no elements, it is called the empty set , or null set , and is denoted by the symbol .∅ For example, the set of digits consists of the collection of numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. If we use the symbol D to denote the set of digits, then we can write D 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 { } = In this notation, the braces { } are used to enclose the objects, or elements , in the set. This method of denoting a set is called the roster method . A second way to denote a set is to use set-builder notation, where the set D of digits is written as D x x is a digit { } = ↑ ↑ ↑ $%''& $%& $'%'& Read as “ D is the set of all x such that x is a digit.” A.1 Algebra Essentials OBJECTIVES 1 Work with Sets (p. A1) 2 Graph Inequalities (p. A4) 3 Find Distance on the Real Number Line (p. A5) 4 Evaluate Algebraic Expressions (p. A6) 5 Determine the Domain of a Variable (p. A7) 6 Use the Laws of Exponents (p. A7) 7 Evaluate Square Roots (p. A9) 8 Use a Calculator to Evaluate Exponents (p. A10) Using Set-builder Notation and the Roster Method (a) E x x is an even digit 0, 2, 4, 6, 8 { } { } = = (b) O x x is an odd digit 1, 3, 5, 7, 9 { } { } = = EXAMPLE 1 Because the elements of a set are distinct, we never repeat elements. For example, we would never write 1, 2, 3, 2 ; { } the correct listing is 1, 2, 3 . { } Because a set is a collection, the order in which the elements are listed is immaterial. 1, 2, 3 , 1, 3, 2 , 2, 1, 3 , { } { } { } and so on, all represent the same set. Review A. 1 Algebra Essentials A. 2 Geometry Essentials A. 3 Polynomials A. 4 Synthetic Division A. 5 Rational Expressions Outline A. 6 Solving Equations A. 7 Complex Numbers; Quadratic Equations in the Complex Number System A. 8 Problem Solving: Interest, Mixture, Uniform Motion, Constant Rate Job Applications A. 9 Interval Notation; Solving Inequalities A. 10 n th Roots; Rational Exponents Appendix A1
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