Number of Functions from an m-element set to an n-element set

To understand the number of functions from an m-element set to an n-element set, let's break down the problem step by step.

1. Definitions:
- An m-element set has m distinct elements.
- An n-element set has n distinct elements.
- A function maps each element from the m-element set to an element in the n-element set.

2. Mapping Elements:
- For each element in the m-element set, there are n possible elements in the n-element set to which it can be mapped.
- Since there are m elements in the m-element set, we have n choices for each element.
- Therefore, the total number of ways to map the elements from the m-element set to the n-element set is n * n * n * ... (m times) = n^m.

3. Example:
- Let's consider an example to illustrate this concept. Suppose we have a 2-element set {a, b} and a 3-element set {x, y, z}.
- For each element in the 2-element set, we have 3 choices in the 3-element set.
- So, the total number of functions from the 2-element set to the 3-element set is 3 * 3 = 9.
- The possible functions can be:
- f(a) = x, f(b) = x
- f(a) = x, f(b) = y
- f(a) = x, f(b) = z
- f(a) = y, f(b) = x
- f(a) = y, f(b) = y
- f(a) = y, f(b) = z
- f(a) = z, f(b) = x
- f(a) = z, f(b) = y
- f(a) = z, f(b) = z

4. Conclusion:
- The number of functions from an m-element set to an n-element set is given by n^m.
- Option 'C' (n^m) is the correct answer.

Therefore, the correct answer is option 'C' (n^m).