Let ` U = { 1,2,3,4,5,6,7,8,9,10} ,A = { 1,3,5,} B = {2,4,6} , C = {4,5,6}`
Find (i) `A^(c) nn B^(c)` (ii) `(A uu B)^(c) nn C^(c)`

Answer

To solve the problem, we need to find the following:

(i) \( A^c \cap B^c \)
(ii) \( (A \cup B)^c \cap C^c \)

Let's break down the solution step by step.

### Step 1: Identify the Universal Set and Given Sets
- Universal Set \( U = \{ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 \} \)
- Set \( A = \{ 1, 3, 5 \} \)
- Set \( B = \{ 2, 4, 6 \} \)
- Set \( C = \{ 4, 5, 6 \} \)

### Step 2: Find the Complements of Sets A, B, and C
- **Complement of A**:
\( A^c = U - A = \{ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 \} - \{ 1, 3, 5 \} = \{ 2, 4, 6, 7, 8, 9, 10 \} \)

- **Complement of B**:
\( B^c = U - B = \{ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 \} - \{ 2, 4, 6 \} = \{ 1, 3, 5, 7, 8, 9, 10 \} \)

- **Complement of C**:
\( C^c = U - C = \{ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 \} - \{ 4, 5, 6 \} = \{ 1, 2, 3, 7, 8, 9, 10 \} \)

### Step 3: Solve Part (i) \( A^c \cap B^c \)
Now we find the intersection of \( A^c \) and \( B^c \):
- \( A^c = \{ 2, 4, 6, 7, 8, 9, 10 \} \)
- \( B^c = \{ 1, 3, 5, 7, 8, 9, 10 \} \)

**Finding the intersection**:
- Common elements in \( A^c \) and \( B^c \) are \( \{ 7, 8, 9, 10 \} \)

Thus,
**Answer for (i)**: \( A^c \cap B^c = \{ 7, 8, 9, 10 \} \)

### Step 4: Solve Part (ii) \( (A \cup B)^c \cap C^c \)
First, we find \( A \cup B \):
- \( A \cup B = \{ 1, 3, 5 \} \cup \{ 2, 4, 6 \} = \{ 1, 2, 3, 4, 5, 6 \} \)

Now find the complement of \( A \cup B \):
- \( (A \cup B)^c = U - (A \cup B) = \{ 1, 2, 3, 4, 5, 6 \}^c = \{ 7, 8, 9, 10 \} \)

Now we find the intersection with \( C^c \):
- \( C^c = \{ 1, 2, 3, 7, 8, 9, 10 \} \)

**Finding the intersection**:
- Common elements in \( (A \cup B)^c \) and \( C^c \) are \( \{ 7, 8, 9, 10 \} \)

Thus,
**Answer for (ii)**: \( (A \cup B)^c \cap C^c = \{ 7, 8, 9, 10 \} \)

### Final Answers
(i) \( A^c \cap B^c = \{ 7, 8, 9, 10 \} \)
(ii) \( (A \cup B)^c \cap C^c = \{ 7, 8, 9, 10 \} \)