Determine the domain and range of the following relations : `{(x,x^(2)):" x is a prime numbers less than 15"}`

Answer

To determine the domain and range of the relation \( R = \{(x, x^2) : x \text{ is a prime number less than 15}\} \), we will follow these steps:

### Step 1: Identify the prime numbers less than 15
The prime numbers less than 15 are:
- 2
- 3
- 5
- 7
- 11
- 13

### Step 2: Form the ordered pairs
For each prime number \( x \), we will form the ordered pair \( (x, x^2) \):
- For \( x = 2 \): \( (2, 2^2) = (2, 4) \)
- For \( x = 3 \): \( (3, 3^2) = (3, 9) \)
- For \( x = 5 \): \( (5, 5^2) = (5, 25) \)
- For \( x = 7 \): \( (7, 7^2) = (7, 49) \)
- For \( x = 11 \): \( (11, 11^2) = (11, 121) \)
- For \( x = 13 \): \( (13, 13^2) = (13, 169) \)

Thus, the relation \( R \) can be expressed as:
\[ R = \{(2, 4), (3, 9), (5, 25), (7, 49), (11, 121), (13, 169)\} \]

### Step 3: Determine the domain
The domain of a relation is the set of all first elements of the ordered pairs. From our ordered pairs, the first elements are:
- 2
- 3
- 5
- 7
- 11
- 13

Thus, the domain \( D \) is:
\[ D = \{2, 3, 5, 7, 11, 13\} \]

### Step 4: Determine the range
The range of a relation is the set of all second elements of the ordered pairs. From our ordered pairs, the second elements are:
- 4
- 9
- 25
- 49
- 121
- 169

Thus, the range \( R \) is:
\[ R = \{4, 9, 25, 49, 121, 169\} \]

### Final Answer:
- **Domain**: \( D = \{2, 3, 5, 7, 11, 13\} \)
- **Range**: \( R = \{4, 9, 25, 49, 121, 169\} \)