If A ={1,2}, then find `AxxAxxA`.

Answer

To find \( A \times A \times A \) where \( A = \{1, 2\} \), we 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 from set \( A \).

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

### Step 3: Form Ordered Triples
We need to form all possible combinations of ordered triples using the elements from set \( A \). Each position in the triple can independently be either \( 1 \) or \( 2 \).

### Step 4: Count the Combinations
Since there are 2 choices (1 or 2) for each of the 3 positions in the ordered triple, the total number of combinations is:
\[ 2 \times 2 \times 2 = 8 \]

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

### Final Result
Thus, the Cartesian product \( A \times A \times A \) is:
\[ A \times A \times A = \{(1, 1, 1), (1, 1, 2), (1, 2, 1), (1, 2, 2), (2, 1, 1), (2, 1, 2), (2, 2, 1), (2, 2, 2)\} \]

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