Number of relations that can be defined on the set A = {a, b, c} is

Options

A. `2^(3)`
B. 6
C. `3^(2)`
D. `2^(9)`

Answer

To find the number of relations that can be defined on the set \( A = \{a, b, c\} \), we can follow these steps:

### Step 1: Determine the number of elements in the set A
The set \( A \) has 3 elements: \( a, b, c \). Therefore, the number of elements \( n \) in set \( A \) is:
\[ n = 3 \]

### Step 2: Calculate the Cartesian product \( A \times A \)
The Cartesian product \( A \times A \) consists of all possible ordered pairs where the first element is from \( A \) and the second element is also from \( A \). The number of elements in \( A \times A \) is given by:
\[ n(A \times A) = n \times n = 3 \times 3 = 9 \]

### Step 3: Determine the number of relations
A relation on a set is defined as a subset of the Cartesian product of the set with itself. The number of subsets of a set with \( m \) elements is given by \( 2^m \). Therefore, the number of relations that can be defined on set \( A \) is:
\[ \text{Number of relations} = 2^{n(A \times A)} = 2^9 \]

### Step 4: Calculate \( 2^9 \)
Now we calculate \( 2^9 \):
\[ 2^9 = 512 \]

### Final Answer
Thus, the number of relations that can be defined on the set \( A = \{a, b, c\} \) is:
\[ \boxed{512} \]

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