Let A and B be two finite sets, then the number of functions from A to B is

Options

A. `n(A).b(B)`
B. `2^(n(A).n(B))`
C. `{n(A)}^(n(B))`
D. `{n(B)}^(n(A))`

Answer

To find the number of functions from one finite set \( A \) to another finite set \( B \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Sets**: Let \( A \) and \( B \) be two finite sets. Denote the number of elements in set \( A \) as \( n(A) \) and the number of elements in set \( B \) as \( n(B) \).

2. **Understanding Functions**: A function from set \( A \) to set \( B \) assigns each element in \( A \) to exactly one element in \( B \).

3. **Choices for Each Element**: For each element in set \( A \), we have \( n(B) \) choices in set \( B \). This means:
- For the first element of \( A \), we can choose any of the \( n(B) \) elements from \( B \).
- For the second element of \( A \), we again have \( n(B) \) choices.
- This continues for all elements in \( A \).

4. **Total Number of Functions**: If \( A \) has \( n(A) \) elements, and each element of \( A \) can independently map to any of the \( n(B) \) elements in \( B \), the total number of functions from \( A \) to \( B \) can be calculated as:
\[ \text{Total functions} = n(B)^{n(A)} \]

5. **Conclusion**: Therefore, the number of functions from set \( A \) to set \( B \) is given by:
\[ n(B)^{n(A)} \]

### Final Answer:
The number of functions from \( A \) to \( B \) is \( n(B)^{n(A)} \).

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