If `A= {-1, 1, 3}` then the number of relations on A is

Options

A. 512
B. 256
C. 1024
D. 128

Answer

To find the number of relations on the set \( A = \{-1, 1, 3\} \), we can follow these steps:

### Step 1: Identify the number of elements in the set \( A \).
The set \( A \) has three elements: \(-1\), \(1\), and \(3\). Therefore, the number of elements \( n \) in set \( A \) is:
\[ n = 3 \]

### Step 2: Use the formula for the number of relations.
The number of relations on a set with \( n \) elements is given by the formula:
\[ \text{Number of relations} = 2^{n^2} \]
This is because a relation on a set can be thought of as a subset of the Cartesian product of the set with itself, which has \( n^2 \) pairs.

### Step 3: Calculate \( n^2 \).
Since \( n = 3 \):
\[ n^2 = 3^2 = 9 \]

### Step 4: Substitute \( n^2 \) into the formula.
Now, substituting \( n^2 \) into the relation formula:
\[ \text{Number of relations} = 2^{9} \]

### Step 5: Calculate \( 2^9 \).
Now we calculate \( 2^9 \):
\[ 2^9 = 512 \]

### Conclusion:
Thus, the number of relations on the set \( A \) is \( 512 \).

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