If A = {5, 7, 8} and B = {9, 11}, then

Options

A. `(9, 5) in A xx B`
B. `(7, 8) in A xx B`
C. `(5, 11) in A xx B`
D. `(9, 11) in A xx B`

Answer

To solve the problem, we need to find the Cartesian product of the sets A and B. The Cartesian product \( A \times B \) is defined as the set of all ordered pairs \( (a, b) \) where \( a \) is an element of set A and \( b \) is an element of set B.

### Step-by-Step Solution:

1. **Identify the sets**:
- Let \( A = \{5, 7, 8\} \)
- Let \( B = \{9, 11\} \)

2. **Write down the Cartesian product definition**:
- The Cartesian product \( A \times B \) consists of all possible ordered pairs \( (a, b) \) where \( a \in A \) and \( b \in B \).

3. **Calculate the ordered pairs**:
- For \( a = 5 \):
- Pair with \( b = 9 \): \( (5, 9) \)
- Pair with \( b = 11 \): \( (5, 11) \)
- For \( a = 7 \):
- Pair with \( b = 9 \): \( (7, 9) \)
- Pair with \( b = 11 \): \( (7, 11) \)
- For \( a = 8 \):
- Pair with \( b = 9 \): \( (8, 9) \)
- Pair with \( b = 11 \): \( (8, 11) \)

4. **List all the ordered pairs**:
- The complete set of ordered pairs is:
\[ A \times B = \{(5, 9), (5, 11), (7, 9), (7, 11), (8, 9), (8, 11)\} \]

5. **Check the options provided**:
- If the options provided in the question are:
- Option A: \( (9, 5) \)
- Option B: \( (7, 8) \)
- Option C: \( (5, 11) \)
- Option D: \( (9, 11) \)

- We can see that:
- Option A \( (9, 5) \) is not in \( A \times B \).
- Option B \( (7, 8) \) is not in \( A \times B \).
- Option C \( (5, 11) \) is in \( A \times B \).
- Option D \( (9, 11) \) is not in \( A \times B \).

6. **Conclusion**:
- The correct option that exists in the Cartesian product \( A \times B \) is **Option C: \( (5, 11) \)**.