2.3 Venn Diagrams, Set Operations, and Data Representation 63 Example 7 Determining Sets from a Venn Diagram Use the Venn diagram in Fig. 2.9 to determine the following sets. Solution a) < > < > A B A C abcdeg abefg a b e g ( ) ( ) {,,, ,,} {,,,,} {, , , } = = b) < > > A B C abcdeg acdg a c d g ( ) {, , , , , } {, , , } {, , , } ′ = = c) > > A B c d f b f g f {, , } {, , } { } ′ ′ = = 7 Now try Exercise 31 U A B I III IV 3 8 # 9 ? % 7 II Figure 2.9 a) U b) A c) ′B d) A B > e) A B < f) ′ A B ( ) < g) n A B ( ) < Solution a) The universal set consists of all the elements within the rectangle, that is, the elements in regions I, II, III, and IV. Thus, U {3, , , , 9, 7, %, ?, #, 8}. ○ = Δ b) Set A consists of the elements in regions I and II. Thus, A {3, , , }. ○ = Δ c) ′B consists of the elements outside set B, or the elements in regions I and IV. Thus, ′ = Δ B {3, , #, 8}. d) A B > consists of the elements that belong to both set A and set B (region II). Thus, A B {, }. > ○ = e) A B < consists of the elements that belong to set A or set B (regions I, II, or III). Thus, A B {3, , , , 9, 7, ?, %}. < ○ = Δ f) ′ A B ( ) < consists of the elements in U that are not in A B < (region IV). Thus, ′ = A( B) {#, 8}. < g) n A B ( ) < represents the number of elements in the union of sets A and B. Thus, = n A B ( ) 8, < as there are eight elements in the union of sets A and B. 7 Now try Exercise 51 Example 8 Using Sets to Organize Data For this Example, we use data from QSR Magazine to form sets of the top fast-food restaurants according to 2022 sales. Let = U the set of the top 10 fast-food restaurants in the U.S., = F the set of the top 5 fast-food restaurants in Florida, and N the = set of the top 5 fast-food restaurants in New York state. U F N {McDonald’s (MD), Starbucks (St), Chick Fil-A (CF), Taco Bell (TB), Burger King (BK), Subway (Su), Wendy’s (We), Dunkin’(Du), Domino’s (Do), Chipotle (Ch)} {CF, We, BK, TB, Ch} {Du, Su, MD, St, BK} = = = Use these sets and the abbreviations given in the universal set to determine the following: a) ′ ′ F N < b) ′ F N ( ) > c) ′ ′ F N > d) ′ F N ( ) < Fabio formaggio/123RF
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