If `A={-1,1}` find `AxxAxxA.`

Explanation

To find the Cartesian product \( A \times A \times A \) for the set \( A = \{-1, 1\} \), we will follow these steps:

### Step 1: Understand the Cartesian Product
The Cartesian product \( A \times A \times A \) means we will create ordered triples (3-tuples) where each element of the triple is taken from the set \( A \).

### Step 2: List the Elements of \( A \)
The set \( A \) contains the elements:
- \( -1 \)
- \( 1 \)

### Step 3: Create Ordered Triples
Since we are forming ordered triples, we will consider all possible combinations of the elements of \( A \) taken three at a time. Each position in the triple can be filled by either \( -1 \) or \( 1 \).

### Step 4: Count the Combinations
Since there are 2 choices for each of the 3 positions in the triple, the total number of ordered triples is:
\[ 2 \times 2 \times 2 = 2^3 = 8 \]

### Step 5: List All Ordered Triples
Now we will list all the combinations:
1. \( (-1, -1, -1) \)
2. \( (-1, -1, 1) \)
3. \( (-1, 1, -1) \)
4. \( (-1, 1, 1) \)
5. \( (1, -1, -1) \)
6. \( (1, -1, 1) \)
7. \( (1, 1, -1) \)
8. \( (1, 1, 1) \)

### Final Result
Thus, the Cartesian product \( A \times A \times A \) is:
\[ A \times A \times A = \{ (-1, -1, -1), (-1, -1, 1), (-1, 1, -1), (-1, 1, 1), (1, -1, -1), (1, -1, 1), (1, 1, -1), (1, 1, 1) \} \]