If A = {1, 2, 3}, B = {3, 4, 5}, then `(A nn B) xx A` is

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

Explanation

To solve the problem, we need to follow these steps:

### Step 1: Identify the Sets
We are given two sets:
- \( A = \{1, 2, 3\} \)
- \( B = \{3, 4, 5\} \)

### Step 2: Find the Intersection of Sets A and B
The intersection of two sets \( A \) and \( B \), denoted as \( A \cap B \), consists of elements that are present in both sets.

- Checking the elements:
- \( 1 \) is not in \( B \)
- \( 2 \) is not in \( B \)
- \( 3 \) is in both \( A \) and \( B \)

Thus, the intersection is:
\[ A \cap B = \{3\} \]

### Step 3: Define the Set C
Let’s define this intersection as set \( C \):
\[ C = A \cap B = \{3\} \]

### Step 4: Find the Cartesian Product \( C \times A \)
The Cartesian product \( C \times A \) consists of all ordered pairs where the first element is from set \( C \) and the second element is from set \( A \).

- Elements of \( C \) are \( \{3\} \)
- Elements of \( A \) are \( \{1, 2, 3\} \)

Now, we can form the pairs:
- Pairing \( 3 \) with each element of \( A \):
- \( (3, 1) \)
- \( (3, 2) \)
- \( (3, 3) \)

Thus, the Cartesian product \( C \times A \) is:
\[ C \times A = \{(3, 1), (3, 2), (3, 3)\} \]

### Final Answer
The result of \( (A \cap B) \times A \) is:
\[ \{(3, 1), (3, 2), (3, 3)\} \]

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