Let n(A) = 6 and n(B) = p . Then , the total number of non - empty relations that can be defined from A to B is

Options

A. `6^(P)`
B. `n^(P+1)`
C. `6p-1`
D. `2^(6p-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 1: Understand the sets
We have two sets:
- Set A with \( n(A) = 6 \) elements.
- Set B with \( n(B) = p \) elements.

### Step 2: Calculate the number of elements in the Cartesian product \( A \times B \)
The number of elements in the Cartesian product \( A \times B \) is given by the formula:
\[ n(A \times B) = n(A) \times n(B) = 6 \times p = 6p \]

### Step 3: Calculate the total number of subsets of \( A \times B \)
The total number of subsets of a set with \( n \) elements is given by \( 2^n \). Therefore, the total number of subsets of the set \( A \times B \) is:
\[ \text{Total subsets} = 2^{n(A \times B)} = 2^{6p} \]

### Step 4: Account for the empty relation
Among all the subsets, one of them is the empty relation. To find the total number of non-empty relations, we need to subtract this empty relation from the total number of subsets:
\[ \text{Total non-empty relations} = 2^{6p} - 1 \]

### Final Answer
Thus, the total number of non-empty relations that can be defined from A to B is:
\[ 2^{6p} - 1 \]