Let n(A) = m and n(B) = n. The total number of non-empty relations that can defined from A to B is

Options

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

Answer

To find the total number of non-empty relations that can be defined from set A to set B, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the number of elements in sets A and B**:
- Let \( n(A) = m \) (the number of elements in set A).
- Let \( n(B) = n \) (the number of elements in set B).

2. **Calculate the number of ordered pairs in the Cartesian product \( A \times B \)**:
- The Cartesian product \( A \times B \) consists of all possible ordered pairs where the first element is from set A and the second element is from set B.
- The total number of ordered pairs in \( A \times B \) is given by:
\[ n(A \times B) = m \times n \]

3. **Determine the total number of relations from A to B**:
- A relation from set A to set B is a subset of the Cartesian product \( A \times B \).
- The total number of subsets of a set with \( k \) elements is \( 2^k \).
- Therefore, the total number of relations from A to B is:
\[ \text{Total Relations} = 2^{m \times n} \]

4. **Calculate the number of non-empty relations**:
- The total number of relations includes the empty relation. To find the number of non-empty relations, we subtract the empty relation from the total number of relations:
\[ \text{Non-empty Relations} = 2^{m \times n} - 1 \]

5. **Final Result**:
- The total number of non-empty relations that can be defined from A to B is:
\[ \text{Non-empty Relations} = 2^{m \times n} - 1 \]

### Summary:
The total number of non-empty relations that can be defined from set A to set B is \( 2^{m \times n} - 1 \).