Consider the following statements.
I. The relation R = `{(x,x^(3))` : x is a parime number less then 10 }
II. The range of the relation
R = { (x + 2, x + 4)), x `in` N , x `lt` 8} is {1,2,3,4,5,6,7}.
Choose the correct option.

Options

A. Only I is true.
B. Only II is true.
C. Both are true
D. Both are false

Answer

To solve the problem, we need to analyze both statements one by one.

### Step 1: Analyze Statement I
The first statement is about the relation \( R = \{(x, x^3) : x \text{ is a prime number less than 10}\} \).

1. Identify the prime numbers less than 10:
- The prime numbers are: 2, 3, 5, and 7.

2. Create ordered pairs \( (x, x^3) \):
- For \( x = 2 \): \( (2, 2^3) = (2, 8) \)
- For \( x = 3 \): \( (3, 3^3) = (3, 27) \)
- For \( x = 5 \): \( (5, 5^3) = (5, 125) \)
- For \( x = 7 \): \( (7, 7^3) = (7, 343) \)

3. Therefore, the relation \( R \) is:
\[ R = \{(2, 8), (3, 27), (5, 125), (7, 343)\} \]

### Step 2: Analyze Statement II
The second statement is about the range of the relation \( R = \{(x + 2, x + 4) : x \in \mathbb{N}, x < 8\} \).

1. Identify the values of \( x \):
- Since \( x \) is a natural number and \( x < 8 \), \( x \) can take values: 1, 2, 3, 4, 5, 6, 7.

2. Create ordered pairs \( (x + 2, x + 4) \):
- For \( x = 1 \): \( (1 + 2, 1 + 4) = (3, 5) \)
- For \( x = 2 \): \( (2 + 2, 2 + 4) = (4, 6) \)
- For \( x = 3 \): \( (3 + 2, 3 + 4) = (5, 7) \)
- For \( x = 4 \): \( (4 + 2, 4 + 4) = (6, 8) \)
- For \( x = 5 \): \( (5 + 2, 5 + 4) = (7, 9) \)
- For \( x = 6 \): \( (6 + 2, 6 + 4) = (8, 10) \)
- For \( x = 7 \): \( (7 + 2, 7 + 4) = (9, 11) \)

3. Collect the second elements to find the range:
- The second elements from the pairs are: 5, 6, 7, 8, 9, 10, 11.
- Thus, the range of the relation is:
\[ \text{Range} = \{5, 6, 7, 8, 9, 10, 11\} \]

### Step 3: Compare the Range with the Given Set
The statement claims that the range is \( \{1, 2, 3, 4, 5, 6, 7\} \). However, we found that the actual range is \( \{5, 6, 7, 8, 9, 10, 11\} \). Therefore, this statement is false.

### Conclusion
Both statements are false. Thus, the correct option is:
- **Option D: Both statements are false.**