Let n(A) = m and n(B) = n, then the number of non-empty relations from A to B is

Options

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

Answer

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

### Step 1: Understand the sizes of the sets
Let \( n(A) = m \) and \( n(B) = n \). This means that set A has \( m \) elements and set B has \( n \) elements.

### Step 2: Calculate the number of ordered pairs
The number of ordered pairs (or the Cartesian product) from set A to set B, denoted as \( A \times B \), is given by the formula:
\[ n(A \times B) = n(A) \times n(B) = m \times n \]
Thus, the total number of ordered pairs from A to B is \( m \times n \).

### Step 3: Calculate the total number of relations
A relation from set A to set B is a subset of the Cartesian product \( A \times B \). The total number of subsets (relations) that can be formed from \( A \times B \) is given by:
\[ \text{Total number of relations} = 2^{n(A \times B)} = 2^{m \times n} \]
This includes all possible subsets, including the empty relation.

### Step 4: Exclude the empty relation
Since we are interested in non-empty relations, we need to subtract the empty relation from the total number of relations. Therefore, the number of non-empty relations from A to B is:
\[ \text{Number of non-empty relations} = 2^{m \times n} - 1 \]

### Final Answer
Thus, the number of non-empty relations from set A to set B is:
\[ 2^{m \times n} - 1 \]

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