`mu = {2,3,4,5,6,7,8} , A = {2,3} , b = {4,5}` and `C = {6,7,8}` , then `A^(c ) nn (B nn c)^(c )` is

Options

A. `A nn C`
B. `A uu B`
C. `B uu C`
D. `A uu C`

Explanation

To solve the problem, we need to find \( A^c \cap (B \cap C)^c \) given the universal set \( \mu \) and the sets \( A \), \( B \), and \( C \).

### Step-by-Step Solution:

1. **Identify the Universal Set and Given Sets:**
- Universal set \( \mu = \{2, 3, 4, 5, 6, 7, 8\} \)
- Set \( A = \{2, 3\} \)
- Set \( B = \{4, 5\} \)
- Set \( C = \{6, 7, 8\} \)

2. **Find the Complement of Set A (\( A^c \)):**
- The complement of \( A \) in \( \mu \) is the set of elements in \( \mu \) that are not in \( A \).
- \( A^c = \mu - A = \{4, 5, 6, 7, 8\} \)

3. **Find the Intersection of Sets B and C (\( B \cap C \)):**
- The intersection \( B \cap C \) consists of elements that are common to both \( B \) and \( C \).
- Since \( B = \{4, 5\} \) and \( C = \{6, 7, 8\} \), there are no common elements.
- \( B \cap C = \emptyset \) (the empty set)

4. **Find the Complement of \( B \cap C \) (\( (B \cap C)^c \)):**
- The complement of the empty set in \( \mu \) is the entire universal set.
- \( (B \cap C)^c = \mu - \emptyset = \{2, 3, 4, 5, 6, 7, 8\} \)

5. **Find the Intersection of \( A^c \) and \( (B \cap C)^c \):**
- Now we need to find \( A^c \cap (B \cap C)^c \).
- We have \( A^c = \{4, 5, 6, 7, 8\} \) and \( (B \cap C)^c = \{2, 3, 4, 5, 6, 7, 8\} \).
- The intersection consists of elements that are in both sets.
- \( A^c \cap (B \cap C)^c = \{4, 5, 6, 7, 8\} \cap \{2, 3, 4, 5, 6, 7, 8\} = \{4, 5, 6, 7, 8\} \)

### Final Answer:
\( A^c \cap (B \cap C)^c = \{4, 5, 6, 7, 8\} \)