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

Answer

To find the Cartesian product \( A \times A \times A \) where \( A = \{1, 2\} \), we will follow these steps:

### Step 1: Understand the Cartesian Product
The Cartesian product \( A \times A \) is the set of all ordered pairs \( (a, b) \) where \( a \) and \( b \) are elements of the set \( A \).

### Step 2: Calculate \( A \times A \)
Since \( A = \{1, 2\} \), we can list the ordered pairs:
- For \( a = 1 \):
- \( (1, 1) \)
- \( (1, 2) \)
- For \( a = 2 \):
- \( (2, 1) \)
- \( (2, 2) \)

Thus, \( A \times A = \{(1, 1), (1, 2), (2, 1), (2, 2)\} \).

### Step 3: Calculate \( A \times A \times A \)
Now we need to find \( A \times A \times A \), which consists of ordered triplets \( (a, b, c) \) where \( a, b, c \) are elements of \( A \).

Using the pairs from \( A \times A \), we can form triplets:
- For \( (1, 1) \):
- \( (1, 1, 1) \)
- \( (1, 1, 2) \)
- For \( (1, 2) \):
- \( (1, 2, 1) \)
- \( (1, 2, 2) \)
- For \( (2, 1) \):
- \( (2, 1, 1) \)
- \( (2, 1, 2) \)
- For \( (2, 2) \):
- \( (2, 2, 1) \)
- \( (2, 2, 2) \)

### Step 4: List All Triplets
Combining all the triplets, we have:
\[ A \times A \times A = \{(1, 1, 1), (1, 1, 2), (1, 2, 1), (1, 2, 2), (2, 1, 1), (2, 1, 2), (2, 2, 1), (2, 2, 2)\} \]

### Step 5: Count the Elements
There are a total of \( 2 \times 2 \times 2 = 8 \) elements in \( A \times A \times A \).

### Final Answer
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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