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

Options

A. {(2, 2), (3, 4), (4, 2), (5, 4)}
B. {(2, 3), (4, 3), (4, 5)}
C. {(2, 4), (3, 4), (4, 4), (4, 5)}
D. {(4, 2), (4, 3), (4, 4), (4, 5)}

Explanation

To solve the problem, we need to find the Cartesian product of the intersection and the union of the given sets A and B.

### Step-by-Step Solution:

1. **Identify the Sets**:
- Set A = {2, 4}
- Set B = {3, 4, 5}

2. **Find the Intersection of A and B (A ∩ B)**:
- The intersection of two sets consists of elements that are common to both sets.
- Common elements in A and B:
- A contains 2 and 4.
- B contains 3, 4, and 5.
- The only common element is 4.
- Therefore, A ∩ B = {4}.

3. **Find the Union of A and B (A ∪ B)**:
- The union of two sets consists of all elements that are in either set, without duplication.
- Elements in A: 2, 4
- Elements in B: 3, 4, 5
- Combining these, we get:
- A ∪ B = {2, 3, 4, 5}.

4. **Find the Cartesian Product of (A ∩ B) and (A ∪ B)**:
- The Cartesian product of two sets X and Y, denoted as X × Y, is the set of all ordered pairs (x, y) where x is in X and y is in Y.
- Here, we need to find {4} × {2, 3, 4, 5}.
- We will pair the element from A ∩ B (which is 4) with each element in A ∪ B:
- (4, 2)
- (4, 3)
- (4, 4)
- (4, 5)

5. **Write the Final Answer**:
- Therefore, the Cartesian product (A ∩ B) × (A ∪ B) = {(4, 2), (4, 3), (4, 4), (4, 5)}.

### Final Result:
The answer is {(4, 2), (4, 3), (4, 4), (4, 5)}.