https://file.aiquickdraw.com/mathbot/file/1725469614266uunmjpzn.jpg

To solve the problem in the image, we need to determine the correct set operation result between the sets \( A \) and \( B \).

### Given:
- \( A = \{x \in \mathbb{N} \mid x = 7k \text{ for some } k \in \mathbb{N}\} \)
- \( B = \{x \in \mathbb{N} \mid x = 2k + 1 \text{ for some } k \in \mathbb{N}, \, 2 \leq k \leq 11 \} \)

### Step 1: Identify elements of \( A \) and \( B \)

- **Set \( A \)**: Since \( x = 7k \), the elements of \( A \) are multiples of 7. Since \( k \) is a natural number, the first few elements of \( A \) are:
  \[
  A = \{7, 14, 21, 28, 35, \dots\}
  \]

- **Set \( B \)**: Since \( x = 2k + 1 \) and \( k \) ranges from 2 to 11, we calculate the values:
  - For \( k = 2 \): \( x = 2(2) + 1 = 5 \)
  - For \( k = 3 \): \( x = 2(3) + 1 = 7 \)
  - For \( k = 4 \): \( x = 2(4) + 1 = 9 \)
  - For \( k = 5 \): \( x = 2(5) + 1 = 11 \)
  - For \( k = 6 \): \( x = 2(6) + 1 = 13 \)
  - For \( k = 7 \): \( x = 2(7) + 1 = 15 \)
  - For \( k = 8 \): \( x = 2(8) + 1 = 17 \)
  - For \( k = 9 \): \( x = 2(9) + 1 = 19 \)
  - For \( k = 10 \): \( x = 2(10) + 1 = 21 \)
  - For \( k = 11 \): \( x = 2(11) + 1 = 23 \)

  So the elements of \( B \) are:
  \[
  B = \{5, 7, 9, 11, 13, 15, 17, 19, 21, 23\}
  \]

### Step 2: Determine the intersection \( A \cap B \)

The intersection of \( A \) and \( B \) includes elements that are present in both sets \( A \) and \( B \). The common elements are:
\[
A \cap B = \{7, 21\}
\]

### Conclusion:

The correct answer is \( A \cap B = \{7, 21\} \), which matches the option that is marked correctly.

Would you like further details or have any questions? Here are some related questions to expand your understanding:

1. How would the intersection change if the range for \( k \) in set \( B \) were different?
2. What would be the intersection if set \( A \) were defined as multiples of 5 instead of 7?
3. How does the range of \( k \) in set \( B \) influence the elements of \( B \)?
4. What is the union of sets \( A \) and \( B \) in this scenario?
5. How would the problem change if set \( A \) was defined with a different formula, like \( x = 5k + 2 \)?

**Tip:** When dealing with set operations, it's helpful to explicitly list the elements of the sets before performing operations like intersections, unions, and differences.

Welche der folgenden Aussagen ist richtig? Seien A = {x ∈ N | x = 7k für ein k ∈ N} und B = {x ∈ N | x = 2k + 1 für ein k ∈ N, 2 ≤ k ≤ 11}.

This problem involves finding the intersection of two sets, A and B, where A = {x ∈ N | x = 7k für ein k ∈ N} and B = {x ∈ N | x = 2k + 1 für ein k ∈ N, 2 ≤ k ≤ 11}. Learn how to identify the correct intersection A ∩ B and understand the underlying concepts in set theory. Suitable for students in Grades 10-12.

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