Certainly! Let's carefully examine the sets \(U\), \(A\), and \(B\) and how to determine their union and intersection.
1. Defining the Sets:
- Universal set (\(U\)): \( U = \{2, 4, 6, 8, 10, 12, 14\} \)
- Set \(A\): \( A = \{2, 6, 9\} \) (Note: The set \{2, 6, (9) 2\} is corrected to \{2, 6, 9\} because sets do not allow duplicates)
- Set \(B\): \( B = \{6, 10, 12, 14\} \)
2. Union of Sets \(A\) and \(B\) (A ∪ B):
- The union of two sets consists of all elements that are in either set or both.
- \(A \cup B = \{x \mid x \in A \text{ or } x \in B\}\)
So, we combine the elements of both sets \(A\) and \(B\):
[tex]\[
A \cup B = \{2, 6, 9\} \cup \{6, 10, 12, 14\} = \{2, 6, 9, 10, 12, 14\}
\][/tex]
3. Intersection of Sets \(A\) and \(B\) (A ∩ B):
- The intersection of two sets consists of all elements that are common to both sets.
- \(A \cap B = \{x \mid x \in A \text{ and } x \in B\}\)
So, we find common elements from both sets \(A\) and \(B\):
[tex]\[
A \cap B = \{2, 6, 9\} \cap \{6, 10, 12, 14\} = \{6\}
\][/tex]
Hence, the results are:
- \( A \cup B = \{2, 6, 9, 10, 12, 14\} \)
- \( A \cap B = \{6\} \)
So we have verified both the union and intersection of the given sets.
