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: Identify the sets
Given:
- Set A = {1, 2, 3}
- Set B = {3, 8}

### Step 2: Find the union of sets A and B
The union of two sets A and B, denoted as \( A \cup B \), is the set of elements that are in A, in B, or in both.

\[ A \cup B = \{1, 2, 3\} \cup \{3, 8\} = \{1, 2, 3, 8\} \]

### Step 3: Find the intersection of sets A and B
The intersection of two sets A and B, denoted as \( A \cap B \), is the set of elements that are in both A and B.

\[ A \cap B = \{1, 2, 3\} \cap \{3, 8\} = \{3\} \]

### Step 4: Find the Cartesian product of \( A \cup B \) and \( A \cap B \)
The Cartesian product of two sets \( X \) and \( Y \), denoted as \( X \times Y \), is the set of all ordered pairs \( (x, y) \) where \( x \) is in \( X \) and \( y \) is in \( Y \).

Now we will calculate \( (A \cup B) \times (A \cap B) \):

\[ (A \cup B) \times (A \cap B) = \{1, 2, 3, 8\} \times \{3\} \]

This means we will pair each element of \( A \cup B \) with the single element in \( A \cap B \):

- 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
\[ (A \cup B) \times (A \cap B) = \{(1, 3), (2, 3), (3, 3), (8, 3)\} \]

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