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The problem states:

Let \( A = \{x : x = 4n + 1, 1 \leq n \leq 5, n \in \mathbb{N} \} \) and \( B = \{y : y = 2n - 3, 2 \leq n \leq 4, n \in \mathbb{N} \} \). The number of relations from \( A \) to \( B \) is a multiple of _______.

### Step-by-step solution:

1. **Set A**:
   - \( A = \{x : x = 4n + 1, 1 \leq n \leq 5, n \in \mathbb{N} \} \).
   - For different values of \(n\), we can calculate:
     - \( n = 1: x = 4(1) + 1 = 5 \)
     - \( n = 2: x = 4(2) + 1 = 9 \)
     - \( n = 3: x = 4(3) + 1 = 13 \)
     - \( n = 4: x = 4(4) + 1 = 17 \)
     - \( n = 5: x = 4(5) + 1 = 21 \)
   - Therefore, \( A = \{5, 9, 13, 17, 21\} \).
   - The cardinality of set \( A \), denoted by \( |A| \), is 5.

2. **Set B**:
   - \( B = \{y : y = 2n - 3, 2 \leq n \leq 4, n \in \mathbb{N} \} \).
   - For different values of \(n\), we can calculate:
     - \( n = 2: y = 2(2) - 3 = 1 \)
     - \( n = 3: y = 2(3) - 3 = 3 \)
     - \( n = 4: y = 2(4) - 3 = 5 \)
   - Therefore, \( B = \{1, 3, 5\} \).
   - The cardinality of set \( B \), denoted by \( |B| \), is 3.

3. **Number of relations**:
   - A relation from \( A \) to \( B \) is a subset of the Cartesian product \( A \times B \). The number of possible relations is the number of subsets of \( A \times B \).
   - The number of elements in \( A \times B \) is \( |A| \times |B| = 5 \times 3 = 15 \).
   - The number of subsets of \( A \times B \) is \( 2^{15} \) (since every element of \( A \times B \) can either be included or excluded from a subset).

4. **Answer**:
   - The number of subsets (and hence the number of relations) is \( 2^{15} = 32768 \).
   - \( 32768 \) is a multiple of \( 32768 \), but also it is a multiple of \( 2^5 = 32 \), \( 2^4 = 16 \), \( 2^3 = 8 \), etc.

Thus, the number of relations is a multiple of **32**.

Would you like more details or have any questions?

### Related questions:
1. How are Cartesian products and relations defined in set theory?
2. How do you calculate the number of subsets of a set with \(n\) elements?
3. What other mathematical problems involve relations between sets?
4. Can the concept of relations be extended to non-finite sets?
5. How is the concept of relations used in computer science, such as in databases?

### Tip:
When working with relations between sets, it's useful to think of them as subsets of the Cartesian product and apply the principles of combinations and power sets.

Let A = {x : x = 4n + 1, 1 ≤ n ≤ 5, n ∈ ℕ} and B = {y : y = 2n - 3, 2 ≤ n ≤ 4, n ∈ ℕ}. The number of relations from A to B is a multiple of ________.

Explore the relations between two sets A and B where set A is defined as {x : x = 4n + 1, 1 ≤ n ≤ 5} and set B as {y : y = 2n - 3, 2 ≤ n ≤ 4}. Learn how to compute the number of possible relations as subsets of Cartesian products, suitable for students in Grades 10-12.

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