Let n(A)=p. Then the number of all relations on A is

Options

A. `2^(p)`
B. `2^(p!)`
C. `2^(p^(2))`
D. None of these

Answer

To find the number of all relations on a set \( A \) where \( n(A) = p \), we can follow these steps:

### Step 1: Understand the Definition of a Relation
A relation from a set \( A \) to a set \( B \) is defined as a subset of the Cartesian product \( A \times B \). In this case, since we are looking for relations on set \( A \) itself, we will consider the Cartesian product \( A \times A \).

### Step 2: Calculate the Size of the Cartesian Product
The size of the Cartesian product \( A \times A \) is given by:
\[ n(A \times A) = n(A) \times n(A) = p \times p = p^2 \]
This means there are \( p^2 \) ordered pairs in \( A \times A \).

### Step 3: Determine the Number of Subsets
Each relation corresponds to a subset of the Cartesian product \( A \times A \). The number of subsets of a set with \( m \) elements is given by \( 2^m \). Therefore, the number of subsets of \( A \times A \) (which has \( p^2 \) elements) is:
\[ 2^{p^2} \]

### Step 4: Conclusion
Thus, the number of all relations on the set \( A \) is:
\[ 2^{p^2} \]

### Final Answer
The number of all relations on \( A \) is \( 2^{p^2} \).

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