If `A={1,2,3}` and `B={3,6,8}`, then `(AcapB)xxA` is equal to

Options

A. `{(1,3),(2,3),(3,3)}`
B. `{(3,1),(3,2),(3,3)}`
C. `{(1,3),(3,1)(3,2)}`
D. None of these

Answer

To solve the problem, we need to find the value of \( (A \cap B) \times A \) where \( A = \{1, 2, 3\} \) and \( B = \{3, 6, 8\} \).

### Step-by-Step Solution:

1. **Identify the Sets**:
We have two sets:
- \( A = \{1, 2, 3\} \)
- \( B = \{3, 6, 8\} \)

2. **Find the Intersection of Sets A and B**:
The intersection \( A \cap B \) consists of elements that are common to both sets.
- The common element in both sets is \( 3 \).
- Therefore, \( A \cap B = \{3\} \).

3. **Calculate the Cartesian Product \( (A \cap B) \times A \)**:
The Cartesian product \( X \times Y \) of two sets \( X \) and \( Y \) is the set of all ordered pairs \( (x, y) \) where \( x \) is in \( X \) and \( y \) is in \( Y \).
- Here, \( (A \cap B) \times A = \{3\} \times \{1, 2, 3\} \).
- We will pair \( 3 \) with each element in \( A \):
- Pairing with \( 1 \): \( (3, 1) \)
- Pairing with \( 2 \): \( (3, 2) \)
- Pairing with \( 3 \): \( (3, 3) \)

4. **Write the Final Result**:
Thus, the Cartesian product \( (A \cap B) \times A \) is:
\[ (A \cap B) \times A = \{(3, 1), (3, 2), (3, 3)\} \]

### Final Answer:
\[ (A \cap B) \times A = \{(3, 1), (3, 2), (3, 3)\} \]