If `A = {1, 2, 3, 6} and B = {2, 4, 6}` be two sets of universal set `U = { 1,2, 3 , 4, 5, 6 , 7}`. Then `( A' cap B')` is equal to

Options

A. `{ 2, 4, 7}`
B. `phi`
C. `{ 5, 7}`
D. `{ 4,6}`

Answer

To solve the problem, we need to find \( A' \cap B' \), where \( A' \) and \( B' \) are the complements of sets \( A \) and \( B \) respectively, with respect to the universal set \( U \).

### Step-by-Step Solution:

1. **Identify the Universal Set \( U \)**:
\[ U = \{ 1, 2, 3, 4, 5, 6, 7 \} \]

2. **Identify the Sets \( A \) and \( B \)**:
\[ A = \{ 1, 2, 3, 6 \} \]
\[ B = \{ 2, 4, 6 \} \]

3. **Find the Complement of Set \( A \) (denoted as \( A' \))**:
\[ A' = U - A = \{ 1, 2, 3, 4, 5, 6, 7 \} - \{ 1, 2, 3, 6 \} \]
- Remove the elements of \( A \) from \( U \):
\[ A' = \{ 4, 5, 7 \} \]

4. **Find the Complement of Set \( B \) (denoted as \( B' \))**:
\[ B' = U - B = \{ 1, 2, 3, 4, 5, 6, 7 \} - \{ 2, 4, 6 \} \]
- Remove the elements of \( B \) from \( U \):
\[ B' = \{ 1, 3, 5, 7 \} \]

5. **Find the Intersection of \( A' \) and \( B' \) (denoted as \( A' \cap B' \))**:
\[ A' \cap B' = \{ 4, 5, 7 \} \cap \{ 1, 3, 5, 7 \} \]
- Identify the common elements:
\[ A' \cap B' = \{ 5, 7 \} \]

Thus, the final answer is:
\[ A' \cap B' = \{ 5, 7 \} \]