Given the sets A = {0, 1, 3}, B = {1, 4, 5}, and C = {0, 1, 3, 4, 5}, what is A ∩ B ∩ C?
Answer (2)
List the elements of sets A, B, and C.
Identify the elements present in all three sets.
The element '1' is the only element common to A, B, and C.
The intersection of A, B, and C is 1 .
Explanation
Understanding the Problem We are given three sets: A = {0, 1, 3}, B = {1, 4, 5}, and C = {0, 1, 3, 4, 5}. The question asks for the intersection of these three sets, denoted as A ∩ B ∩ C, which represents the elements that are common to all three sets.
Listing the Elements To find the intersection, we need to identify the elements that are present in all three sets A, B, and C. Let's list the elements of each set:
Set A: {0, 1, 3} Set B: {1, 4, 5} Set C: {0, 1, 3, 4, 5}
Identifying Common Elements Now, let's find the elements that are common to all three sets:
The element '0' is in A and C, but not in B.
The element '1' is in A, B, and C.
The element '3' is in A and C, but not in B.
The element '4' is in B and C, but not in A.
The element '5' is in B and C, but not in A.
Therefore, the only element that is present in all three sets is '1'.
Finding the Intersection The intersection of the sets A, B, and C is the set containing the elements common to all three sets. In this case, the intersection is {1}. So, A ∩ B ∩ C = {1}.
Examples
Understanding set intersections is crucial in database management. For instance, if you have three databases of customers (A, B, and C), finding the intersection (A ∩ B ∩ C) helps identify customers present in all three databases. This is useful for targeted marketing campaigns or identifying common users across different platforms, ensuring efficient resource allocation and personalized user experiences.
The intersection of the sets A, B, and C is {1}, as '1' is the only element common to all three sets. All other elements do not appear in all three sets. Therefore, A ∩ B ∩ C = {1}.
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