Lecture Review: Venn Diagrams and Problem Solving

Venn Diagram Terminology Refresher

  • "At most three": Refers to a quantity of 1, 2, or 3 items.
  • "At least two" (with three items total): Refers to a quantity of 2 or 3 items.

Travel Agency Problem (Problem 3)

This problem involves a travel agency with potential customers wanting to visit Hawaii, Disney, or Las Vegas.

  • Total Potential Customers: 125

  • Innermost Intersection (Hawaii, Disney, and Las Vegas):

    • The intersection of all three destinations is given as 18 customers.

  • Specific Intersections and Totals:

    • Las Vegas and Disney Intersection: A total of 26 customers. The lecture notes that the values 16, 5, and 8 were derived, using the 18 from the triple intersection. (Implied: The region for only Las Vegas and Disney, without Hawaii, would be 26 - 18 = 8 customers).
    • Disney Total: 47 customers want to go to Disney.
      • Existing regions already identified within the Disney circle were 5, 18, and 8 (possibly Hawai'i and Disney only, All three, and Las Vegas and Disney only, respectively, based on filling a Venn diagram).
      • Sum of these known regions in Disney: 5 + 18 + 8 = 31 customers.
      • The number of customers who want to go to Disney only is calculated as the total Disney customers minus the sum of the known shared regions: 47 - 31 = 16 customers.
      • Note on discrepancy: The lecturer also stated, after adding 16, 18, 8 (which sums to 42), then subtracting 39, that the value 29 "belongs in there" (referring to the Disney-only region). This numerical calculation differs from the logical derivation of 16. We are using the consistently derived result of 16.

  • Total Sum (within Venn Diagram):

    • After filling in various sections, the sum of all known regions within the Venn diagram was 103 customers.
    • Customers wanting none of the three destinations: To find customers who do not want to travel to any of these three places, subtract the sum of those in the regions from the total potential customers: 125 - 103 = 22 customers.

  • Question 1: How many want to travel to Disney or Las Vegas, but not Hawaii?

    • The lecturer provided the numbers 5, 16, 18, and 8 for this calculation, summing to 5 + 16 + 18 + 8 = 47 customers.
    • Note on ambiguity: The sum of 47 represents the entire Disney circle. However, the condition "but not Hawaii" explicitly requires excluding regions within the Hawaii set, which would include the 18 (Hawaii, Disney, Las Vegas) and potentially the 5 (Hawaii and Disney only, if that's what it represents). This suggests a potential contradiction between the question's condition and the given sum.

  • Question 2: How many went at most to two places?

    • This phrase means "two places or less". To calculate this, one would sum the number of customers who went to 0, 1, or 2 places.

US Cities Survey Problem

This problem surveys 33 US cities regarding the presence of a professional sports team, a symphony, or a children's museum.

  • Total Surveyed Cities: 33

  • Filling the Venn Diagram (from center outward):

    • Cities with all three (Sports, Symphony, and Children's Museum) (S ext{ and } Y ext{ and } M): 5 cities.
    • Symphony and Children's Museum only ((Y ext{ and } M) ext{ not } S): The total intersection of Symphony and Children's Museum is 9. Subtracting the triple intersection: 9 - 5 = 4 cities.
    • Sports and Children's Museum only ((S ext{ and } M) ext{ not } Y): The total intersection of Sports and Children's Museum is 7. Subtracting the triple intersection: 7 - 5 = 2 cities.
    • Sports and Symphony only ((S ext{ and } Y) ext{ not } M): The total intersection of Sports and Symphony is 11. Subtracting the triple intersection: 11 - 5 = 6 cities.

  • Calculating "Only One" Regions:

    • Children's Museum only (M ext{ not } (S ext{ or } Y)$): The total for Children's Museum is 15. Subtracting the relevant shared regions: 15 - (5 + 4 + 2) = 15 - 11 = 4 cities.
    • Symphony only (Y ext{ not } (S ext{ or } M)$): The total for Symphony is 17. Subtracting the relevant shared regions: 17 - (5 + 4 + 6) = 17 - 15 = 2 cities.
    • Sports Team only (S ext{ not } (Y ext{ or } M)$): The total for Sports Team is 16. Subtracting the relevant shared regions: 16 - (5 + 2 + 6) = 16 - 13 = 3 cities.

  • Sum of all regions within the Venn Diagram:

    • Adding all calculated regions: (Sports only) 3 + (Symphony only) 2 + (Children's Museum only) 4 + (S & Y only) 6 + (S & M only) 2 + (Y & M only) 4 + (S & Y & M) 5 = 26 cities.

  • Cities with none of the three classifications (outside diagram):

    • Subtract the sum of cities within the Venn diagram from the total surveyed cities: 33 - 26 = 7 cities.

  • Final Question: How many cities surveyed had only one professional (type of organization)?

    • This requires summing the "only one" regions:
      • (Sports only) + (Symphony only) + (Children's Museum only)
      • 3 + 2 + 4 = 9$$ cities.

  • Practice Test Clarification:

    • The practice test is solely for practice and will not be graded by the instructor.