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.