Let A be a finite set containing n distinct elements. The number of functions that can be defined from A to A is

Options

A. `2^(n)`
B. `n^(!)`
C. `n^(2)`
D. `n^(n)`

Answer

To solve the problem of finding the number of functions that can be defined from a finite set \( A \) containing \( n \) distinct elements to itself, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
- We have a set \( A \) with \( n \) distinct elements.
- We want to find the number of functions \( f: A \to A \).

2. **Identifying Domain and Co-domain**:
- The domain of the function is the set \( A \) which has \( n \) elements.
- The co-domain of the function is also the set \( A \) which has \( n \) elements.

3. **Choosing Outputs for Each Input**:
- For each element in the domain (which has \( n \) elements), we need to choose an output from the co-domain (which also has \( n \) elements).
- Each element in the domain can map to any of the \( n \) elements in the co-domain.

4. **Calculating the Total Number of Functions**:
- For the first element in the domain, there are \( n \) choices for its image in the co-domain.
- For the second element in the domain, there are again \( n \) choices.
- This continues for all \( n \) elements in the domain.
- Therefore, the total number of functions is given by multiplying the number of choices for each element:
\[ n \times n \times n \times \ldots \text{ (n times)} = n^n \]

5. **Conclusion**:
- Thus, the number of functions that can be defined from \( A \) to \( A \) is \( n^n \).

### Final Answer:
The number of functions that can be defined from \( A \) to \( A \) is \( n^n \).

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