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
The set \( A \) has 3 elements: \( a, b, c \).

### Step 2: Understand the concept of relations
A relation from set \( A \) to itself can be defined as a subset of the Cartesian product \( A \times A \). The Cartesian product \( A \times A \) consists of all ordered pairs where the first element is from set \( A \) and the second element is also from set \( A \).

### Step 3: Calculate the Cartesian product
The Cartesian product \( A \times A \) will have \( n \times n \) pairs, where \( n \) is the number of elements in set \( A \). Since \( A \) has 3 elements:
\[ A \times A = 3 \times 3 = 9 \]
Thus, there are 9 ordered pairs in \( A \times A \):
\[ A \times A = \{(a,a), (a,b), (a,c), (b,a), (b,b), (b,c), (c,a), (c,b), (c,c)\} \]

### Step 4: Determine the number of subsets
Each relation corresponds to a subset of \( A \times A \). The number of subsets of a set with \( m \) elements is given by \( 2^m \). Since we have 9 ordered pairs:
\[ \text{Number of relations} = 2^{|A \times A|} = 2^9 \]

### Step 5: Calculate \( 2^9 \)
Calculating \( 2^9 \):
\[ 2^9 = 512 \]

### Conclusion
The number of relations that can be defined on the set \( A = \{a, b, c\} \) is \( 512 \).

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