If A `{1,2}, B = {1,3}, " then " (A xx B)cup (B xx A)` is equal to

Options

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

Answer

To solve the problem, we need to find the union of the Cartesian products \( A \times B \) and \( B \times A \). Let's break this down step by step.

### Step 1: Identify the sets
We have:
- \( A = \{1, 2\} \)
- \( B = \{1, 3\} \)

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

Calculating \( A \times B \):
- Pairing \( 1 \) from \( A \) with each element in \( B \):
- \( (1, 1) \)
- \( (1, 3) \)

- Pairing \( 2 \) from \( A \) with each element in \( B \):
- \( (2, 1) \)
- \( (2, 3) \)

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

### Step 3: Calculate \( B \times A \)
Next, we calculate \( B \times A \), which consists of all ordered pairs where the first element is from set \( B \) and the second element is from set \( A \).

Calculating \( B \times A \):
- Pairing \( 1 \) from \( B \) with each element in \( A \):
- \( (1, 1) \)
- \( (1, 2) \)

- Pairing \( 3 \) from \( B \) with each element in \( A \):
- \( (3, 1) \)
- \( (3, 2) \)

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

### Step 4: Calculate the union \( (A \times B) \cup (B \times A) \)
Now we need to find the union of the two sets calculated above.

Combining \( A \times B \) and \( B \times A \):
\[ (A \times B) \cup (B \times A) = \{(1, 1), (1, 3), (2, 1), (2, 3)\} \cup \{(1, 1), (1, 2), (3, 1), (3, 2)\} \]

### Step 5: List all unique elements
When we take the union, we combine the elements and remove duplicates:
- From \( A \times B \): \( (1, 1), (1, 3), (2, 1), (2, 3) \)
- From \( B \times A \): \( (1, 1), (1, 2), (3, 1), (3, 2) \)

Combining these, we get:
\[ \{(1, 1), (1, 3), (2, 1), (2, 3), (1, 2), (3, 1), (3, 2)\} \]

### Final Answer
Thus, the final answer is:
\[ (A \times B) \cup (B \times A) = \{(1, 1), (1, 2), (1, 3), (2, 1), (2, 3), (3, 1), (3, 2)\} \]

---