Let A be a finite set. The number of relations on A where A has 3 elements are : (i) 9 (ii) 6 (iii) 256 (iv) 512

Options

A. 9
B. 6
C. 256
D. 512

Answer

To find the number of relations on a finite set \( A \) with 3 elements, we can follow these steps:

### Step 1: Understand the Definition of a Relation
A relation on a set \( A \) is a subset of the Cartesian product \( A \times A \). If \( A \) has \( n \) elements, then \( A \times A \) will have \( n^2 \) elements.

### Step 2: Determine the Size of the Set
Given that the set \( A \) has 3 elements, we can denote the elements of \( A \) as \( A = \{ a_1, a_2, a_3 \} \).

### Step 3: Calculate the Size of the Cartesian Product
The Cartesian product \( A \times A \) will have \( n^2 = 3^2 = 9 \) elements. Thus, \( A \times A = \{ (a_1, a_1), (a_1, a_2), (a_1, a_3), (a_2, a_1), (a_2, a_2), (a_2, a_3), (a_3, a_1), (a_3, a_2), (a_3, a_3) \} \).

### Step 4: Calculate the Number of Subsets
The number of relations on the set \( A \) is equal to the number of subsets of \( A \times A \). Since there are 9 elements in \( A \times A \), the number of subsets (and hence the number of relations) is given by \( 2^{n^2} \).

### Step 5: Substitute the Value of \( n \)
Substituting \( n = 3 \):
\[ \text{Number of relations} = 2^{3^2} = 2^9 \]

### Step 6: Calculate \( 2^9 \)
Calculating \( 2^9 \):
\[ 2^9 = 512 \]

### Conclusion
Thus, the number of relations on a set \( A \) with 3 elements is \( 512 \).

### Final Answer
The correct option is (iv) 512.

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