At an auto show, a car dealer asked the first 10 visitors their ages and the kinds of cars they own. The Venn diagram shows the details.
A Venn diagram titled Ages of Car Owners, indicating the ages of owners of each type of car. Each of the three circles in the Venn diagram represents a type of car and the three circles are labeled Sedans (Set X), Sports Cars (Set Z), and SUVs (Set Y). The ages of owners of only Set X are 28, 27, and 41, the age of an owner of only set Z is 34, and the ages of owners of only Set Y are 45, 38, and 51. The ages of owners of all three types of cars is left blank. The age of an owner of Set X and Set Z is 42. The age of an owner of Set Z and Set Y is 49. The age of an owner of Set X and Set Y is 39
Based on the diagram, which of the following represents the ages of car owners in the set
(X ∩ ~Y) ∪ (Z ∩ Y)?
To determine the ages of car owners in the set (X ∩ ~Y) ∪ (Z ∩ Y), we need to look at the overlapping regions in the Venn diagram.
1. (X ∩ ~Y) represents the intersection of Set X and the complement of Set Y. This means it includes the ages of owners who own sedans (Set X) but do not own SUVs (complement of Set Y). The ages in this intersection are 28, 27, and 41.
2. (Z ∩ Y) represents the intersection of Set Z and Set Y. This means it includes the ages of owners who own sports cars (Set Z) and SUVs (Set Y). The ages in this intersection are 49.
Now, we need to take the union of these two intersections:
(X ∩ ~Y) ∪ (Z ∩ Y) = {28, 27, 41} ∪ {49}
Therefore, the ages of car owners in the set (X ∩ ~Y) ∪ (Z ∩ Y) are 28, 27, 41, and 49.
To find the ages of car owners in the set (X ∩ ~Y) ∪ (Z ∩ Y), we need to understand the given Venn diagram and apply the correct operations.
Let's break down the given information and assign variables:
- Sedans (Set X): Ages of owners of only Set X are 28, 27, and 41. The age of an owner of Set X and Set Z is 42. The age of an owner of Set X and Set Y is 39.
- Sports Cars (Set Z): The age of an owner of only Set Z is 34. The age of an owner of Set X and Set Z is 42. The age of an owner of Set Z and Set Y is 49.
- SUVs (Set Y): The ages of owners of only Set Y are 45, 38, and 51. The age of an owner of Set Z and Set Y is 49. The age of an owner of Set X and Set Y is 39.
Based on this information, we can now determine the ages of car owners in the set (X ∩ ~Y) ∪ (Z ∩ Y).
To determine (X ∩ ~Y), we need to find the ages of owners who own Sedans (Set X) but do not own SUVs (Set Y).
- Considering the information given, the ages of owners who own both Set X and Set Y is 39. This means the ages of owners in (X ∩ Y).
- Therefore, (X ∩ ~Y) would exclude this age, so we need to remove 39 from (X ∩ Y).
To determine (Z ∩ Y), we need to find the ages of owners who own Sports Cars (Set Z) and SUVs (Set Y).
- Considering the information given, the age of an owner of Set Z and Set Y is 49. This age belongs to both Set Z and Set Y and should be included in (Z ∩ Y).
Now, let's calculate (X ∩ ~Y) ∪ (Z ∩ Y) by combining the ages described above:
- (X ∩ ~Y) = (28, 27, 41) - (39) = (28, 27, 41)
- (Z ∩ Y) = (49)
So, (X ∩ ~Y) ∪ (Z ∩ Y) = (28, 27, 41) ∪ (49)
Hence, the ages of car owners in the set (X ∩ ~Y) ∪ (Z ∩ Y) are 28, 27, 41, and 49.