If A={1,2} and B={2,3} , then show that:
`AxxB ne B xxA`

Explanation

To show that \( A \times B \neq B \times A \) for the sets \( A = \{1, 2\} \) and \( B = \{2, 3\} \), we will first define the Cartesian products \( A \times B \) and \( B \times A \), and then demonstrate that they are not equal by listing their elements.

### Step 1: Define the sets
Let \( A = \{1, 2\} \) and \( B = \{2, 3\} \).

### Step 2: Calculate \( A \times B \)
The Cartesian product \( A \times B \) consists of all ordered pairs \( (a, b) \) where \( a \in A \) and \( b \in B \).

- For \( a = 1 \):
- Pair with \( b = 2 \): \( (1, 2) \)
- Pair with \( b = 3 \): \( (1, 3) \)

- For \( a = 2 \):
- Pair with \( b = 2 \): \( (2, 2) \)
- Pair with \( b = 3 \): \( (2, 3) \)

Thus,
\[ A \times B = \{(1, 2), (1, 3), (2, 2), (2, 3)\} \]

### Step 3: Calculate \( B \times A \)
The Cartesian product \( B \times A \) consists of all ordered pairs \( (b, a) \) where \( b \in B \) and \( a \in A \).

- For \( b = 2 \):
- Pair with \( a = 1 \): \( (2, 1) \)
- Pair with \( a = 2 \): \( (2, 2) \)

- For \( b = 3 \):
- Pair with \( a = 1 \): \( (3, 1) \)
- Pair with \( a = 2 \): \( (3, 2) \)

Thus,
\[ B \times A = \{(2, 1), (2, 2), (3, 1), (3, 2)\} \]

### Step 4: Compare \( A \times B \) and \( B \times A \)
Now we compare the two sets:
- \( A \times B = \{(1, 2), (1, 3), (2, 2), (2, 3)\} \)
- \( B \times A = \{(2, 1), (2, 2), (3, 1), (3, 2)\} \)

Since the ordered pairs in \( A \times B \) and \( B \times A \) are different, we conclude that:
\[ A \times B \neq B \times A \]

### Conclusion
Thus, we have shown that \( A \times B \neq B \times A \).

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