Let n(A)=m and n(B)=n Then, the total number of non-empty relations that can be defined from
A to B is:

Options

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

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 1: Understand the concept of 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 ∈ A and b ∈ B.

### Step 2: Determine the sizes of sets A and B
Let n(A) = m and n(B) = n. This means that set A has m elements and set B has n elements.

### Step 3: Calculate the size of the Cartesian product A × B
The number of elements in the Cartesian product A × B is given by:
\[ n(A \times B) = n(A) \times n(B) = m \times n \]

### Step 4: Calculate the total number of subsets of A × B
The total number of subsets of a set with k elements is given by \( 2^k \). Therefore, the total number of subsets of A × B is:
\[ 2^{m \times n} \]

### Step 5: Exclude the empty relation
Among all the subsets of A × B, one of them is the empty set (which represents the empty relation). Therefore, to find the total number of non-empty relations, we subtract 1 from the total number of subsets:
\[ \text{Total non-empty relations} = 2^{m \times n} - 1 \]

### Final Answer
Thus, the total number of non-empty relations that can be defined from set A to set B is:
\[ 2^{mn} - 1 \]

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