If A = {1, 2, 3} and B = {3, 8}, then `(A uu B) xx (A nn B)`is

Options

A. {(3, 1), (3, 2), (3, 3), (3, 8)}
B. {(1, 3), (2, 3), (3, 3), (8, 3)}
C. {(1, 2), (2, 2), (3, 3), (8, 8)}
D. {(8, 3), (8, 2), (8, 1), (8, 8)}

Explanation

To solve the problem, we need to find the Cartesian product of the union and intersection of two sets A and B. Let's break it down step by step.

### Step 1: Define the Sets
Given:
- \( A = \{1, 2, 3\} \)
- \( B = \{3, 8\} \)

### Step 2: Find the Union of Sets A and B
The union of two sets combines all unique elements from both sets.
\[ A \cup B = \{1, 2, 3\} \cup \{3, 8\} \]
Here, we take all elements from both sets without repetition:
\[ A \cup B = \{1, 2, 3, 8\} \]

### Step 3: Find the Intersection of Sets A and B
The intersection of two sets includes only the elements that are present in both sets.
\[ A \cap B = \{1, 2, 3\} \cap \{3, 8\} \]
The common element is:
\[ A \cap B = \{3\} \]

### Step 4: Find the Cartesian Product of \( A \cup B \) and \( A \cap B \)
The Cartesian product of two sets \( X \) and \( Y \) is the set of all ordered pairs \( (x, y) \) where \( x \) is from set \( X \) and \( y \) is from set \( Y \).
\[ (A \cup B) \times (A \cap B) = \{1, 2, 3, 8\} \times \{3\} \]
We will create pairs by taking each element from \( A \cup B \) and pairing it with the element from \( A \cap B \):
- Pairing 1 with 3: \( (1, 3) \)
- Pairing 2 with 3: \( (2, 3) \)
- Pairing 3 with 3: \( (3, 3) \)
- Pairing 8 with 3: \( (8, 3) \)

Thus, the Cartesian product is:
\[ (A \cup B) \times (A \cap B) = \{(1, 3), (2, 3), (3, 3), (8, 3)\} \]

### Final Answer
The result of \( (A \cup B) \times (A \cap B) \) is:
\[ \{(1, 3), (2, 3), (3, 3), (8, 3)\} \]