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

Answer

To find \( A \times A \times A \) where \( A = \{-1, 1\} \), we will follow these steps:

### Step 1: Calculate \( A \times A \)

The Cartesian product \( A \times A \) consists of all ordered pairs where the first element is from the first set \( A \) and the second element is from the second set \( A \).

\[ A \times A = \{(-1, -1), (-1, 1), (1, -1), (1, 1)\} \]

### Step 2: Calculate \( A \times A \times A \)

Now, we need to find \( A \times A \times A \). This means we will take the result from \( A \times A \) and pair each of those ordered pairs with each element of \( A \).

1. Pair \( (-1, -1) \) with each element of \( A \):
- \( (-1, -1, -1) \)
- \( (-1, -1, 1) \)

2. Pair \( (-1, 1) \) with each element of \( A \):
- \( (-1, 1, -1) \)
- \( (-1, 1, 1) \)

3. Pair \( (1, -1) \) with each element of \( A \):
- \( (1, -1, -1) \)
- \( (1, -1, 1) \)

4. Pair \( (1, 1) \) with each element of \( A \):
- \( (1, 1, -1) \)
- \( (1, 1, 1) \)

Combining all these, we get:

\[ A \times A \times A = \{(-1, -1, -1), (-1, -1, 1), (-1, 1, -1), (-1, 1, 1), (1, -1, -1), (1, -1, 1), (1, 1, -1), (1, 1, 1)\} \]

### Final Answer

Thus, the result of \( A \times A \times A \) is:

\[ A \times A \times A = \{(-1, -1, -1), (-1, -1, 1), (-1, 1, -1), (-1, 1, 1), (1, -1, -1), (1, -1, 1), (1, 1, -1), (1, 1, 1)\} \]