If `A={0,1,3}`, then the number of relations on A is____________.

Answer

To find the number of relations on the set \( A = \{0, 1, 3\} \), we can follow these steps:

### Step 1: Understand the Concept of Relations
A relation on a set \( A \) is a subset of the Cartesian product \( A \times A \). The Cartesian product \( A \times A \) consists of all possible ordered pairs where the first element is from set \( A \) and the second element is also from set \( A \).

### Step 2: Determine the Size of the Set
The set \( A \) has 3 elements: \( 0, 1, \) and \( 3 \).

### Step 3: Calculate the Cartesian Product
The Cartesian product \( A \times A \) will have \( n^2 \) pairs, where \( n \) is the number of elements in set \( A \). Since \( A \) has 3 elements, we calculate:
\[ |A \times A| = 3 \times 3 = 9 \]
Thus, the set \( A \times A \) consists of the following pairs:
\[ A \times A = \{(0,0), (0,1), (0,3), (1,0), (1,1), (1,3), (3,0), (3,1), (3,3)\} \]

### Step 4: Determine the Number of Subsets
A relation is any subset of \( A \times A \). The number of subsets of a set with \( m \) elements is given by \( 2^m \). Since \( A \times A \) has 9 elements, the number of subsets (relations) is:
\[ \text{Number of relations} = 2^{|A \times A|} = 2^9 \]

### Step 5: Calculate the Final Answer
Calculating \( 2^9 \):
\[ 2^9 = 512 \]
Thus, the number of relations on the set \( A \) is \( 512 \).

### Final Answer
The number of relations on \( A \) is **512**.

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