Answers 1. {s, v, w, x, y, z } 2. {s, v, w} 3. { } 4. 1, 4) 5. (, 4) ´ 9, ) b. A ¨ B {a, c, e} The intersection of A and B includes only those elements that are common to both sets. c. A ¨ C { } (the empty set) Because A and C share no common elements, the intersection of A and C is the empty set (also called the null set). Skill Practice Given: A {r, s, t, u, v,w} B {s, v, w, y, z} C {x, y, z} Find: 1. 2. 3. B ´ C A ¨ B A ¨ C Given the sets: A {x 0 x 3} B {x 0 x 2} C {x 0 x 5} Graph the following sets. Then express each set in interval notation. a. A ¨ B b. A ´ C Solution: It is helpful to visualize the graphs of individual sets on the number line before taking the union or intersection. ( Finding the Union and Intersection of Sets 65 4 3 2 1 0 1 2 3 4 5 6 6 5 4 3 2 1 0 1 2 3 4 5 6 ( 6 5 4 3 2 1 0 1 2 3 4 5 6 A C includes all elements from set A along with the elements from set C. a. Graph of A {x 0 x 3} Graph of B {x 0 x 2} Graph of A ¨ B (the “overlap”) Interval notation: 2, 3) Note that the set A ¨ B represents the real numbers greater than or equal to 2 and less than 3. This relationship can be written more concisely as a compound inequality: 2 x 3. We can interpret this inequality as “x is between 2 and 3, including x 2.” b. Graph of A {x 0 x 3} Graph of C {x 0 x 5} Graph of A ´ C Interval notation: ( , 3) ´ 5, ) Skill Practice Given the sets: and determine the union or intersection and express the answer in C 5x 0 x 96, interval notation. 4. A ¨ B 5. B ´ C A 5x 0 x 16, B 5x 0 x 6 46, Example 2 TIP: The empty set may be denoted by the symbol { } or by the symbol . ( 6 54 3 2 1 0 1 2 3 4 5 6 6 5 4 3 2 1 0 1 2 3 4 5 6 ( 6 54 3 2 1 0 1 2 3 4 5 6 86 Chapter 1 Linear Equations and Inequalities in One Variable In Example 3, we find the union and intersection of sets expressed in interval notation.
miller_intermediate_algebra_4e_ch1_3
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