62 CHAPTER 2 Sets Using set-builder notation, the union of set A and set B is indicated by = | ∈ ∈ A B x x A x B { or }. < The three shaded regions of Fig. 2.8, regions I, II, and III, together represent the union of sets A and B. If an element is common to both sets, it is listed only once in the union of the sets. U A B I II III IV A < B Figure 2.8 Example 5 The Union 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< Solution a) The union of set A and set B is the set of elements that are in set A or set B or in both sets. Therefore, = = A B a c e g i abcde abcdegi {,,,,} {,,, ,} {,,, ,,,}. < < b) The union of set B and set C is the set of elements that are in set B or set C or in both sets. Therefore, = = B C a b c d e b d f h j a b c d e f h j {,,, ,} {, ,,,} {,,, ,,,,}. < < c) The union of set A and set C is the set of elements that are in set A or set C or in both sets. Therefore, = = A C a c e g i b d f h j abcdef ghij {,,,,} {, ,,,} {,,, ,,,,,,}. < < Note that = A C U. < 7 Now try Exercise 29 To help understand the distinction between intersection and union, compare Example 3 to Example 5. We used the same sets in both examples, but in Example 3 we determined the intersection of sets and in Example 5 we determined the union of sets. Example 6 Union, Intersection, and Complement Given U a b c d e f g A a b e g B a c d e C b e f {, , , , , , } {, , , } {, , , } {, , } = = = = determine each of the following. a) A B A C ( ) ( ) < > < b) ′ A B C ( ) < > c) ′ ′ A B > Union of Two Sets
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