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Solving Word Problems with Venn Diagrams: Applications of Set Theory

This section explores the use of Venn diagrams to solve word problems involving sets, particularly in customer preference scenarios. We demonstrate how to set up a three-set Venn diagram to illustrate relationships among different groups, such as preferences for Hawaii, Las Vegas, and Disney World. By filling in the diagram from the inside out, we can effectively answer questions related to customer desires, such as how many did not wish to travel to any of the listed destinations. Examples clarify the process, making it easier to visualize and derive solutions.

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Solving Word Problems with Venn Diagrams: Applications of Set Theory

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  1. Section 2.5 Applications of Sets

  2. Applications = Word Problems • We will solve word problems using Venn Diagrams. • We will set up the Venn Diagram for the problem. • We then answer any questions about the sets in the problem.

  3. The 3-Set Venn Diagram U A B II III I V IV VI C VII VIII

  4. Order of Filling in the Regions: Inside-Out 1) Fill in information for Region V 2) Fill in information for Regions II, IV, VI 3) Fill in information for Regions I, III, VII 4) Fill in information for Region VIII

  5. Page 85, Example 3 • 125 Customers • 68 Hawaii • 53 Las Vegas • 47 Disney World • 34 Hawaii and Las Vegas • 26 Las Vegas and Disney World • 23 Hawaii and Disney World • 18 all three

  6. Example 3 Venn Diagram L U H D

  7. Work upwards from bottom to top for the descriptions in Example 3 on pg 85 • 18 wished to travel to all three destinations. • Fill this in first, and work upwards in the list of descriptions.

  8. L+D = 26. 26-18 = 8 L U H 18 5 8 D

  9. H+L = 34. 34-18 = 16 L U H 16 18 5 8 D

  10. D = 47. 47-18-8-5 = 16 L U H 16 18 5 8 16 D

  11. L = 53. 53-18-16-8 = 11 L U H 11 16 18 5 8 16 D

  12. H = 68. 68-18-16-5 = 29 L U H 11 29 16 18 5 8 16 D

  13. Now use the Venn Diagram to answer the questions • Page 85 questions. How many surveyed a) did not wish to travel to any of these destinations? Looking at the Venn Diagram, we see this is 22.

  14. None of the destinations = 22

  15. b) wish to travel only to Hawaii?

  16. c) Travel to Disney World and Las Vegas, but not Hawaii

  17. Travel to Disney World or Las Vegas, but not Hawaii = 35

  18. Wished to travel to exactly one of the destinations = 56

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