If A={1,2},B ={1,3} then `(A xxB) cup (BxxA)` 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 go through the steps systematically.

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

### Step 2: Find \( A \times B \)
The Cartesian product \( A \times B \) consists of all ordered pairs \( (x, y) \) where \( x \) is from set \( A \) and \( y \) 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: Find \( B \times A \)
The Cartesian product \( B \times A \) consists of all ordered pairs \( (x, y) \) where \( x \) is from set \( B \) and \( y \) 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: Find the Union \( (A \times B) \cup (B \times A) \)
Now we need to find the union of the two sets obtained in the previous steps.

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

Listing all unique elements:
- \( (1, 1) \) (already counted)
- \( (1, 3) \)
- \( (2, 1) \)
- \( (2, 3) \)
- \( (1, 2) \)
- \( (3, 1) \)
- \( (3, 2) \)

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

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