If A={1,2} and B={x,y}, then find `AXB`. Also find the number of its subsets.

Answer

To solve the problem, we need to find the Cartesian product of the sets A and B, denoted as \( A \times B \), and then determine the number of subsets of that Cartesian product.

**Step 1: Identify the sets A and B.**
- Given \( A = \{1, 2\} \)
- Given \( B = \{x, y\} \)

**Step 2: Find the Cartesian product \( A \times B \).**
The Cartesian product \( A \times B \) consists of all possible ordered pairs where the first element is from set A and the second element is from set B.

We can list the ordered pairs as follows:
- From the first element of A (which is 1):
- Pairing with the first element of B (x): \( (1, x) \)
- Pairing with the second element of B (y): \( (1, y) \)

- From the second element of A (which is 2):
- Pairing with the first element of B (x): \( (2, x) \)
- Pairing with the second element of B (y): \( (2, y) \)

Thus, the Cartesian product \( A \times B \) is:
\[ A \times B = \{(1, x), (1, y), (2, x), (2, y)\} \]

**Step 3: Count the number of elements in \( A \times B \).**
The set \( A \times B \) has 4 elements: \( (1, x), (1, y), (2, x), (2, y) \).

**Step 4: Find the number of subsets of \( A \times B \).**
The number of subsets of a set is given by \( 2^n \), where \( n \) is the number of elements in the set.

Since \( A \times B \) has 4 elements, the number of subsets is:
\[ 2^4 = 16 \]

**Final Answer:**
- The Cartesian product \( A \times B = \{(1, x), (1, y), (2, x), (2, y)\} \)
- The number of subsets of \( A \times B \) is 16.

---