Let A and B be two finite sets having m and n elements respectively. Then the total number of mappings from A to B is

Options

A. mn
B. `2^(mn)`
C. `m^(n)`
D. `n^(m)`

Answer

To find the total number of mappings (or functions) from set A to set B, where set A has m elements and set B has n elements, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Sets**:
- Let set A = {a1, a2, a3, ..., am} with m elements.
- Let set B = {b1, b2, b3, ..., bn} with n elements.

2. **Mapping Definition**:
- A mapping (or function) from set A to set B assigns each element of A to exactly one element of B.

3. **Choices for Each Element**:
- For the first element of A (let's say a1), there are n possible choices in set B (it can map to any of the n elements).
- For the second element of A (a2), there are also n choices in set B.
- This continues for all m elements in set A.

4. **Total Choices Calculation**:
- Since each of the m elements in A can independently map to any of the n elements in B, the total number of mappings can be calculated by multiplying the number of choices for each element.
- Therefore, the total number of mappings is given by:
\[ \text{Total Mappings} = n \times n \times n \times \ldots \text{(m times)} = n^m \]

5. **Conclusion**:
- The total number of mappings from set A to set B is \( n^m \).

### Final Answer:
The total number of mappings from A to B is \( n^m \).

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