If `A={1,3,5,7)` and `B={2,4,6}`, find `AxxA`

Answer

To find \( A \times A \) where \( A = \{1, 3, 5, 7\} \), we will follow the definition of the Cartesian product of a set with itself. The Cartesian product \( A \times A \) consists of all possible ordered pairs \( (a, b) \) where \( a \) and \( b \) are elements of the set \( A \).

### Step-by-Step Solution:

1. **Identify the Elements of Set A**:
- The elements of set \( A \) are \( 1, 3, 5, 7 \).

2. **Form Ordered Pairs**:
- We will create ordered pairs by taking each element of \( A \) as the first element of the pair and pairing it with every element of \( A \) as the second element.

3. **Create Pairs for Each Element**:
- For the first element \( 1 \):
- Pairs: \( (1, 1), (1, 3), (1, 5), (1, 7) \)
- For the second element \( 3 \):
- Pairs: \( (3, 1), (3, 3), (3, 5), (3, 7) \)
- For the third element \( 5 \):
- Pairs: \( (5, 1), (5, 3), (5, 5), (5, 7) \)
- For the fourth element \( 7 \):
- Pairs: \( (7, 1), (7, 3), (7, 5), (7, 7) \)

4. **Combine All Ordered Pairs**:
- Now, we combine all the pairs we created:
\[ A \times A = \{(1, 1), (1, 3), (1, 5), (1, 7), (3, 1), (3, 3), (3, 5), (3, 7), (5, 1), (5, 3), (5, 5), (5, 7), (7, 1), (7, 3), (7, 5), (7, 7)\} \]

### Final Answer:
\[ A \times A = \{(1, 1), (1, 3), (1, 5), (1, 7), (3, 1), (3, 3), (3, 5), (3, 7), (5, 1), (5, 3), (5, 5), (5, 7), (7, 1), (7, 3), (7, 5), (7, 7)\} \]