If A = {x,y,z} B = {a,x} ; then find `(AxxB) `.

Answer

To find the Cartesian product \( A \times B \) where \( A = \{x, y, z\} \) and \( B = \{a, x\} \), we will follow these steps:

### Step 1: Understand the Cartesian Product
The Cartesian product \( A \times B \) is defined as the set of all ordered pairs \( (a, b) \) where \( a \) is an element from set \( A \) and \( b \) is an element from set \( B \).

### Step 2: List the Elements of Set A and Set B
- Set \( A = \{x, y, z\} \)
- Set \( B = \{a, x\} \)

### Step 3: Form Ordered Pairs
We will form ordered pairs by taking each element from set \( A \) and pairing it with each element from set \( B \).

1. Pair \( x \) from \( A \) with each element in \( B \):
- \( (x, a) \)
- \( (x, x) \)

2. Pair \( y \) from \( A \) with each element in \( B \):
- \( (y, a) \)
- \( (y, x) \)

3. Pair \( z \) from \( A \) with each element in \( B \):
- \( (z, a) \)
- \( (z, x) \)

### Step 4: Combine All Ordered Pairs
Now, we combine all the ordered pairs we formed:
\[ A \times B = \{(x, a), (x, x), (y, a), (y, x), (z, a), (z, x)\} \]

### Final Answer
Thus, the Cartesian product \( A \times B \) is:
\[ A \times B = \{(x, a), (x, x), (y, a), (y, x), (z, a), (z, x)\} \]

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