The Cartesian product `A xxA` has 9 elements among which are found `(-1," "0)" "a n d" "(0," "1)` . Find the set A and the remaining elements of `A xxA` .

Explanation

To solve the problem, we need to find the set \( A \) and the remaining elements of the Cartesian product \( A \times A \).

### Step-by-Step Solution:

1. **Understanding the Cartesian Product**:
The Cartesian product \( A \times A \) consists of all ordered pairs \( (a, b) \) where \( a \) and \( b \) are elements of set \( A \). The number of elements in \( A \times A \) is given by \( n(A) \times n(A) \), where \( n(A) \) is the number of elements in set \( A \).

2. **Given Information**:
We know that \( A \times A \) has 9 elements. Therefore, we can set up the equation:
\[ n(A) \times n(A) = 9 \]
This implies:
\[ n(A)^2 = 9 \]
Taking the square root of both sides, we find:
\[ n(A) = 3 \]

3. **Identifying Elements of Set \( A \)**:
We are given two elements from the Cartesian product: \( (-1, 0) \) and \( (0, 1) \). Since these pairs are formed from the elements of set \( A \), we can conclude that the elements of \( A \) must include \( -1 \), \( 0 \), and \( 1 \).

Thus, we can write:
\[ A = \{-1, 0, 1\} \]

4. **Finding Remaining Elements of \( A \times A \)**:
Now, we need to find all the elements of \( A \times A \). The Cartesian product will include all combinations of the elements in \( A \):
- Pairing \( -1 \) with each element in \( A \):
- \( (-1, -1) \)
- \( (-1, 0) \) (given)
- \( (-1, 1) \)

- Pairing \( 0 \) with each element in \( A \):
- \( (0, -1) \)
- \( (0, 0) \)
- \( (0, 1) \) (given)

- Pairing \( 1 \) with each element in \( A \):
- \( (1, -1) \)
- \( (1, 0) \)
- \( (1, 1) \)

5. **Listing All Elements**:
Now, we can list all the elements of \( A \times A \):
\[ A \times A = \{ (-1, -1), (-1, 0), (-1, 1), (0, -1), (0, 0), (0, 1), (1, -1), (1, 0), (1, 1) \} \]

The given elements \( (-1, 0) \) and \( (0, 1) \) are already included in this list.

### Remaining Elements:
The remaining elements of \( A \times A \) that were not given in the question are:
- \( (-1, -1) \)
- \( (-1, 1) \)
- \( (0, -1) \)
- \( (0, 0) \)
- \( (1, -1) \)
- \( (1, 0) \)
- \( (1, 1) \)

### Final Answer:
Set \( A = \{-1, 0, 1\} \) and the remaining elements of \( A \times A \) are:
- \( (-1, -1) \)
- \( (-1, 1) \)
- \( (0, -1) \)
- \( (0, 0) \)
- \( (1, -1) \)
- \( (1, 0) \)
- \( (1, 1) \)