Section 2.5 Application of Sets

1. Section 2.5Application of Sets

2. What You Will Learn • Application of Sets

3. Example 1: Yogurt Taste Test A yogurt company wishes to introduce a new yogurt flavor. The company is considering two flavors: raspberry cheesecake (R) and orange creme (O). In a survey of 250 people it was found that 180 people liked raspberry cheesecake. 139 people liked orange creme. 82 people liked both flavors.

4. Example 1: Yogurt Taste Test Of those surveyed, how many people a) did not like either raspberry cheesecake or orange crème? b) liked raspberry cheesecake, but not orange creme? c) liked orange creme, but not raspberry cheesecake? d) liked either raspberry cheesecake or orange creme?

5. Example 1: Yogurt Taste Test • Solution People surveyed is 250: n(U) = 250. People surveyed who liked raspberry cheesecake is 180: n(R) = 180. People surveyed who liked orange creme is 139: n(O) = 139. People surveyed who liked both raspberry cheesecake and orange creme is 82: n(R ⋂ O) = 82.

6. Example 1: Yogurt Taste Test • Solution Region II is n(R ⋂ O) = 82 I is n(R) – n(R ⋂ O) = 180 – 82 = 98 III is n(O) – n(R ⋂ O) = 139 – 98 = 57 Add I, II, & III 98 + 82 + 57 = 237 IV isn(U) – n(R ⋃ O) = 250 – 237 = 13

7. Try this: P. 80 # 2

8. Applications Involving 3 Sets • The procedure to work problems with three sets is generally the same. Start by completing region V. Next complete regions II, IV, and VI. Then complete regions I, III, and VII. Finally, complete region VIII. When you are constructing Venn diagrams, be sure to check your work carefully.

9. Example 3: Travel Packages Liberty Travel surveyed 125 potential customers. The following information was obtained. 68 wished to travel to Hawaii. 53 wished to travel to Las Vegas. 47 wished to travel to Disney World. 34 wished to travel to Hawaii and Las Vegas. 26 wished to travel to Las Vegas and Disney World. 23 wished to travel to Hawaii and Disney World. 18 wished to travel to all three destinations.

10. Example 3: Travel Packages Use a Venn diagram to answer the following questions. How many of those surveyed a) did not wish to travel to any of these destinations? b) wished to travel only to Hawaii? c) wished to travel to Disney World and Las Vegas, but not to Hawaii? d) wished to travel to Disney World or Las Vegas, but not to Hawaii? e) wished to travel to exactly one of these destinations?

11. Example 3: Travel Packages Solution V is all 3 destinations = 18 II is Hawaii & Disney = 23 – 18 = 5 IV is Hawaii & Vegas = 34 – 18 = 16 VI is Dis. & Vegas = 26 – 18 = 8 I is Hawaii only = 68 – 5 – 18 – 16 = 29

12. Example 3: Travel Packages Solution III is Disney only = 47 – 5 – 18 – 8 = 16 VII is Vegas only = 53 – 16 – 18 – 8 = 11 Add I – VII = 103 VIII = 125 – 103 = 22

13. Example 3: Travel Packages Solution a) 22 did not wish to travel to any of these destinations b) 29 wished to travel to Hawaii only c) 8 wished to travel to Disney and Vegas but not Hawaii

14. Example 3: Travel Packages Solution d) 16 + 8 + 11 = 35 wished to travel to Disney or Vegas but not Hawaii e) 29 + 16 + 11 = 56 wished to to travel to exactly one of these destinations

15. Try this: p. 80 #5

16. Homework P. 80 # 1- 12, skip 2 and 5