If `R` is a relation from a finite set A having `m` elements to a finite set B having `n` elements then the number of relations from A to B is

Options

A. A. `2^(m n)`
B. B. `2^(m n)-1`
C. C. `n^m`
D. D. `m^n`

Answer

To solve the problem of finding the number of relations from a finite set A with m elements to a finite set B with n elements, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Relations**: A relation from set A to set B is a subset of the Cartesian product A × B. The Cartesian product A × B consists of all possible ordered pairs (a, b) where a is an element from set A and b is an element from set B.

2. **Calculating the Size of the Cartesian Product**:
- Since set A has m elements and set B has n elements, the total number of ordered pairs in A × B is given by:
\[ |A \times B| = m \times n \]

3. **Finding the Number of Subsets**:
- A relation is simply a subset of the Cartesian product A × B. The number of subsets of a set with k elements is given by \(2^k\).
- Therefore, the number of subsets of A × B, which corresponds to the number of relations from A to B, is:
\[ \text{Number of relations} = 2^{|A \times B|} = 2^{m \times n} \]

4. **Conclusion**:
- Thus, the total number of relations from set A to set B is:
\[ 2^{mn} \]

5. **Final Answer**:
- The correct option is **A: \(2^{mn}\)**.