To solve the given problem, let's go through the required steps to prove the union and intersection of the sets \( A \) and \( B \).
Given:
- \( U = \{2, 4, 6, 8, 10, 12, 14\} \)
- \( A = \{2, 6, 10, 12\} \)
- \( B = \{6, 10, 12, 14\} \)
### Finding the Union \((A \cup B)\):
The union of two sets \(A\) and \(B\) is the set of elements that are in either \( A \) or \( B \) or in both.
To compute \( A \cup B \), we combine all unique elements from both sets \(A\) and \(B\):
[tex]\[
A \cup B = \{2, 6, 10, 12\} \cup \{6, 10, 12, 14\}
\][/tex]
Combine all the elements:
[tex]\[
A \cup B = \{2, 6, 10, 12, 14\}
\][/tex]
### Finding the Intersection \((A \cap B)\):
The intersection of two sets \(A\) and \(B\) is the set of elements that are in both \(A\) and \(B\).
To compute \( A \cap B \), we identify all elements that are common to both sets \(A\) and \(B\):
[tex]\[
A \cap B = \{2, 6, 10, 12\} \cap \{6, 10, 12, 14\}
\][/tex]
Identify the common elements:
[tex]\[
A \cap B = \{6, 10, 12\}
\][/tex]
### Summary
Thus, the union and intersection of sets \( A \) and \( B \) are:
[tex]\[
A \cup B = \{2, 6, 10, 12, 14\}
\][/tex]
[tex]\[
A \cap B = \{6, 10, 12\}
\][/tex]
This concludes our solution.
