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

Answer

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 \( A \times A \times A \) consists of all possible ordered triples \( (a, b, c) \) where \( a, b, c \) are elements of the set \( A \).

### Step 2: Identify the Elements of Set A
The set \( A \) has the elements:
- \( -1 \)
- \( 1 \)

### Step 3: Generate All Possible Ordered Triples
We will combine each element of \( A \) with every other element of \( A \) to form ordered triples.

1. Start with \( -1 \):
- \( (-1, -1, -1) \)
- \( (-1, -1, 1) \)
- \( (-1, 1, -1) \)
- \( (-1, 1, 1) \)

2. Now, use \( 1 \):
- \( (1, -1, -1) \)
- \( (1, -1, 1) \)
- \( (1, 1, -1) \)
- \( (1, 1, 1) \)

### Step 4: List All Ordered Triples
Combining all the ordered triples we generated, we have:
\[ A \times A \times A = \{ (-1, -1, -1), (-1, -1, 1), (-1, 1, -1), (-1, 1, 1), (1, -1, -1), (1, -1, 1), (1, 1, -1), (1, 1, 1) \} \]

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

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