If `AxxB={(a,x),(a,y),(b,x),(b,y)}`, then find A and B

Explanation

To solve the problem, we need to identify the sets \( A \) and \( B \) from the given Cartesian product \( A \times B = \{(a,x), (a,y), (b,x), (b,y)\} \).

### Step-by-Step Solution:

1. **Understanding the Cartesian Product**:
The Cartesian product \( A \times B \) consists of all possible ordered pairs \( (p, q) \) where \( p \) is an element from set \( A \) and \( q \) is an element from set \( B \).

2. **Identifying Elements in the Ordered Pairs**:
From the given set \( A \times B \), we can see the ordered pairs:
- \( (a, x) \)
- \( (a, y) \)
- \( (b, x) \)
- \( (b, y) \)

3. **Extracting Elements for Set \( A \)**:
The first elements of the ordered pairs are \( a \) and \( b \). Therefore, the set \( A \) can be defined as:
\[ A = \{a, b\} \]

4. **Extracting Elements for Set \( B \)**:
The second elements of the ordered pairs are \( x \) and \( y \). Therefore, the set \( B \) can be defined as:
\[ B = \{x, y\} \]

5. **Final Result**:
Thus, the sets \( A \) and \( B \) are:
\[ A = \{a, b\}, \quad B = \{x, y\} \]