If U= {1,2,3,4,5,6,7,8,9},
A= {2,4,6,8},
B {2,3,5,7} . Verify that :
(i) `(AuuB)=AcapB`
`.

Answer

To verify the statement \( A \cup B = A \cap B \), we will first find the union and intersection of the sets \( A \) and \( B \).

**Step 1: Identify the sets.**
- Given:
- Universal set \( U = \{1, 2, 3, 4, 5, 6, 7, 8, 9\} \)
- Set \( A = \{2, 4, 6, 8\} \)
- Set \( B = \{2, 3, 5, 7\} \)

**Step 2: Find the union of sets \( A \) and \( B \).**
- The union of two sets \( A \) and \( B \) is the set of elements that are in either \( A \), \( B \), or both.
- Therefore, \( A \cup B = \{2, 4, 6, 8\} \cup \{2, 3, 5, 7\} \).
- Combining the elements, we get:
\[ A \cup B = \{2, 3, 4, 5, 6, 7, 8\} \]

**Step 3: Find the intersection of sets \( A \) and \( B \).**
- The intersection of two sets \( A \) and \( B \) is the set of elements that are in both \( A \) and \( B \).
- Therefore, \( A \cap B = \{2, 4, 6, 8\} \cap \{2, 3, 5, 7\} \).
- The common element is:
\[ A \cap B = \{2\} \]

**Step 4: Compare the results of union and intersection.**
- We found:
- \( A \cup B = \{2, 3, 4, 5, 6, 7, 8\} \)
- \( A \cap B = \{2\} \)
- Clearly, \( A \cup B \neq A \cap B \).

**Conclusion:**
- Therefore, we have verified that \( A \cup B \) is not equal to \( A \cap B \).

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