2. Let [tex]A = \{1,3,5\}[/tex] and [tex]B = \{2,3,4\}[/tex].
(a) Find [tex]A \cup B, A \cap B, A-B[/tex], and [tex]B-A[/tex].
(b) Find the complement of [tex]A[/tex] with respect to the universal set [tex]U = \{1,2,3,4,5,6\}[/tex].
Answer (2)
The results of the set operations are: A ∪ B = {1, 2, 3, 4, 5}, A ∩ B = {3}, A - B = {1, 5}, B - A = {2, 4}, and the complement of A is {2, 4, 6}.
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A ∪ B combines elements from both sets: A ∪ B = { 1 , 2 , 3 , 4 , 5 } .
A ∩ B identifies common elements: A ∩ B = { 3 } .
A − B finds elements in A but not in B: A − B = { 1 , 5 } .
B − A finds elements in B but not in A: B − A = { 2 , 4 } .
A c (complement of A) includes elements in U but not in A: A c = { 2 , 4 , 6 } .
A ∪ B = { 1 , 2 , 3 , 4 , 5 } , A ∩ B = { 3 } , A − B = { 1 , 5 } , B − A = { 2 , 4 } , A c = { 2 , 4 , 6 }
Explanation
Understanding the Problem We are given two sets, A = { 1 , 3 , 5 } and B = { 2 , 3 , 4 } , and a universal set U = { 1 , 2 , 3 , 4 , 5 , 6 } . We need to find the union of A and B ( A ∪ B ), the intersection of A and B ( A ∩ B ), the set difference A − B , the set difference B − A , and the complement of A with respect to U ( A c ).
Finding the Union The union of two sets, A ∪ B , is the set of all elements that are in A , in B , or in both. In this case, A = { 1 , 3 , 5 } and B = { 2 , 3 , 4 } , so A ∪ B = { 1 , 2 , 3 , 4 , 5 } .
Finding the Intersection The intersection of two sets, A ∩ B , is the set of all elements that are in both A and B . In this case, A = { 1 , 3 , 5 } and B = { 2 , 3 , 4 } , so A ∩ B = { 3 } .
Finding the Set Difference A - B The set difference A − B is the set of all elements that are in A but not in B . In this case, A = { 1 , 3 , 5 } and B = { 2 , 3 , 4 } , so A − B = { 1 , 5 } .
Finding the Set Difference B - A The set difference B − A is the set of all elements that are in B but not in A . In this case, A = { 1 , 3 , 5 } and B = { 2 , 3 , 4 } , so B − A = { 2 , 4 } .
Finding the Complement of A The complement of A with respect to U , denoted A c , is the set of all elements that are in U but not in A . In this case, A = { 1 , 3 , 5 } and U = { 1 , 2 , 3 , 4 , 5 , 6 } , so A c = { 2 , 4 , 6 } .
Final Answer Therefore, we have: A ∪ B = { 1 , 2 , 3 , 4 , 5 } A ∩ B = { 3 } A − B = { 1 , 5 } B − A = { 2 , 4 } A c = { 2 , 4 , 6 }
Examples
Understanding sets and set operations is fundamental in computer science, particularly in database management and data analysis. For example, if you have two lists of customers, one who purchased product A and another who purchased product B, you can use set operations to find customers who bought both (intersection), customers who bought either (union), or customers who bought only one of the products (set difference). These operations help businesses target their marketing efforts more effectively.
