60 CHAPTER 2 Sets Intersection The word intersection brings to mind the area common to two crossing streets. The red car in the figure is in the intersection of the two streets. The set operation intersection is defined as follows. Definition: Intersection The intersection of sets A and B, symbolized by A B, > is the set containing all the elements that are common to both set A and set B. Using set-builder notation, the intersection of set A and set B is indicated by = | ∈ ∈ A B x x A x B { and }. > The shaded region, region II, in Fig. 2.6 represents the intersection of sets A and B. Example 2 Venn Diagram of Two Sets For the sets U A, , and B, construct a Venn diagram illustrating the relationship between set A and set B. = = = U A B {1, 2, 4, 5, 6, 7, 8, 9, 10} {1, 2, 3, 4, 5} {2, 4, 6, 8, 10} Solution First, determine the intersection of sets A and B. The elements 2 and 4 are common to both sets. Therefore, = A B {2, 4} > and we place these elements in region II of Fig. 2.7. Next, we complete region I by determining the elements in set A that have not been placed in region II. Therefore, the elements 1, 3, and 5 are placed in region I. Next, we complete region III by determining the elements in set B that have not been placed in region II. Therefore, the elements 6, 8, and 10 are placed in region III. Finally, note that the elements 7 and 9 are in the universal set, but are not in set A and are not in set B. We place these elements in region IV. Fig. 2.7 shows the completed Venn diagram. 7 Now try Exercise 13 I II III A B U IV 1 3 5 2 4 7 9 6 8 10 Figure 2.7 Example 3 The Intersection of Sets Given U abcdef ghij A a c e g i B a b c d e C b d f h j {,,, ,,,,,,} {,,,,} {, , , , } {, , , ,} = = = = Determine a) A B. > b) B C. > c) A C. > U A B I II III IV Figure 2.6 We next work an example that involves three sets.
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