2.3 Venn Diagrams, Set Operations, and Data Representation 67 *See Instructor Answer Appendix Example 12 The Cartesian Product of Two Sets Given = A {orange,banana,apple} and = B {1, 2}, determine the following. a) × A B b) × B A c) × A A d) × B B Solution a) × = A B {(orange, 1), (orange, 2), (banana, 1), (banana, 2), (apple, 1), (apple, 2)} b) × = B A {(1, orange), (1, banana), (1, apple), (2, orange), (2, banana), (2, apple)} c) × = A A {(orange, orange), (orange, banana), (orange, apple), (banana, orange), (banana, banana), (banana, apple), (apple, orange), (apple, banana), (apple, apple)} d) × = B B {(1, 1), (1, 2), (2, 1), (2, 2)} 7 Now try Exercise 55 Instructor Resources for Section 2.3 in MyLab Math • Objective-Level Videos 2.3 • Animation: Union of Two Sets • PowerPoint Lecture Slides 2.3 • MyLab Exercises and Assignments 2.3 We can see from Example 12 that, in general, × ≠ × A B B A. The ordered pairs in × A B are not the same as the ordered pairs in × B A. For example ≠ (orange, 1) (1, orange). In general, if a set A has m elements and a set B has n elements, then the number of ordered pairs in × A B will be × m n. In Example 12, set A contains 3 elements and set B contains 2 elements. Notice that × A B contains × 3 2, or 6, ordered pairs. Exercises Warm Up Exercises In exercises 1–8, fill in the blank with an appropriate word, phrase, or symbol(s). 1. The set of all the elements in the universal set that are not in set A is called the _________ of set A. Complement 2. The set containing all the elements that are members of set A or of set B or of both sets is called the _________ of set A and set B. Union 3. The set containing all the elements that are common to both set A and set B is called the _________ of set A and set B. Intersection 4. The set of elements that belong to set A, but not to set B, is called the _________ of two sets A and B. Difference 5. The set of all possible ordered pairs of the form a b (, ), where ∈ a A and ∈ b B, is called the _________ product of set A and set B. Cartesian 6. Two sets with no elements in common are called _________ sets. Disjoint 7. If set A has m elements and set B has n elements, the Cartesian product × A B has _________ elements. × m n 8. In a Venn diagram with two overlapping sets there are _________ regions. Four Practice the Skills For Exercises 9–10, given the universal set, U, and set A, a) determine the complement, ′A, and b) illustrate the relationship among set U A, , and ′A in a Venn diagram. 9. = = U A {1, 2, 3, 4, 5, 6}, {1, 3, 5} a) A {2, 4, 6} ′ = b) * 10. = = U abcdef ghij A aei {,,, ,,,,,,}, { , , } a) A b c d f g h j {,, ,,,,} ′ = b) * For Exercises 11–14, for sets U, A, and B, construct a Venn diagram and place the elements in the proper region. 11. U A B {1, 2, 3, 4, 5, 6, 7, 8} {1, 6, 8} {2, 4, 6, 7, 8} * = = = 12. U a b c d e f A a c B b e f {, , , , , } {, } {, , } * = = = 13. U abcdef ghijk A a b e f h j B b c d f j {, , , , , , , , , , } {,,,,,} {,, ,,} * = = = 14. U A B {, , , , , , , , } {, , , , } {, , , } * = ΓΔΘΛΠΣΦΨΩ = Γ Δ Θ Λ Π = Λ Π Σ Φ SECTION 2.3
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