Let U= {1, 2, 3, 4, 5, 6, 7, 8,9}, A={1, 2, 3, 4}, B = {2, 4, 6, 8) and C={3, 4, 5, 6}. Find
`( A uu B)'`

Answer

To solve the problem, we need to find the complement of the union of sets A and B with respect to the universal set U. Let's break this down step by step.

### Step 1: Identify the Universal Set and the Sets A and B
- Universal Set \( U = \{1, 2, 3, 4, 5, 6, 7, 8, 9\} \)
- Set \( A = \{1, 2, 3, 4\} \)
- Set \( B = \{2, 4, 6, 8\} \)

### Step 2: Find the Union of Sets A and B
The union of two sets includes all elements that are in either set A or set B (or in both).

\[ A \cup B = \{1, 2, 3, 4\} \cup \{2, 4, 6, 8\} \]

To find \( A \cup B \), we combine the elements:
- From A: 1, 2, 3, 4
- From B: 2, 4, 6, 8

Now, combine these while removing duplicates:
\[ A \cup B = \{1, 2, 3, 4, 6, 8\} \]

### Step 3: Find the Complement of \( A \cup B \)
The complement of a set with respect to the universal set consists of all elements in the universal set that are not in the set.

To find the complement \( (A \cup B)' \):
\[ (A \cup B)' = U - (A \cup B) \]

Substituting the values:
\[ (A \cup B)' = \{1, 2, 3, 4, 5, 6, 7, 8, 9\} - \{1, 2, 3, 4, 6, 8\} \]

### Step 4: Perform the Set Subtraction
Now, we subtract the elements of \( A \cup B \) from the universal set \( U \):
- Start with \( U \): 1, 2, 3, 4, 5, 6, 7, 8, 9
- Remove elements: 1, 2, 3, 4, 6, 8

The remaining elements are:
\[ (A \cup B)' = \{5, 7, 9\} \]

### Final Answer
Thus, the complement of the union of sets A and B is:
\[ (A \cup B)' = \{5, 7, 9\} \]

---