Let R be a relation on D defined by
`R={(1+x,1+x^2):x le 5, x in N}`
Which of the following is false ?

Options

A. R={(2,2),(3,5),(4,10),(5,17),(6,26)}
B. Domain of R ={2,3,4,5,6}
C. Range of R={2,3,4,5,6}
D. None of the above

Answer

To solve the problem, we need to analyze the relation \( R \) defined by:

\[ R = \{(1+x, 1+x^2) : x \leq 5, x \in \mathbb{N}\} \]

### Step 1: Determine the values of \( x \)
Since \( x \) is a natural number and \( x \leq 5 \), the possible values of \( x \) are \( 1, 2, 3, 4, 5 \).

### Step 2: Calculate the pairs in the relation \( R \)
We will substitute each value of \( x \) into the relation:

- For \( x = 1 \):
\[ (1+1, 1+1^2) = (2, 2) \]

- For \( x = 2 \):
\[ (1+2, 1+2^2) = (3, 5) \]

- For \( x = 3 \):
\[ (1+3, 1+3^2) = (4, 10) \]

- For \( x = 4 \):
\[ (1+4, 1+4^2) = (5, 17) \]

- For \( x = 5 \):
\[ (1+5, 1+5^2) = (6, 26) \]

### Step 3: Compile the relation \( R \)
Now we can compile the pairs we calculated:
\[ R = \{(2, 2), (3, 5), (4, 10), (5, 17), (6, 26)\} \]

### Step 4: Identify the domain and range
- **Domain**: The first elements of each pair:
\[ \text{Domain} = \{2, 3, 4, 5, 6\} \]

- **Range**: The second elements of each pair:
\[ \text{Range} = \{2, 5, 10, 17, 26\} \]

### Step 5: Analyze the options
Now we need to compare the domain and range with the provided options to identify which statement is false.

Assuming the options are:
1. Domain is \( \{2, 3, 4, 5, 6\} \)
2. Range is \( \{2, 5, 10, 17, 26\} \)
3. Domain is \( \{2, 3, 4, 5\} \)
4. Range is \( \{2, 5, 10, 17\} \)

### Conclusion
- The domain we calculated is \( \{2, 3, 4, 5, 6\} \) which matches option 1.
- The range we calculated is \( \{2, 5, 10, 17, 26\} \) which matches option 2.
- Option 3 states the domain is \( \{2, 3, 4, 5\} \), which is false because we have \( 6 \) in the domain.
- Option 4 states the range is \( \{2, 5, 10, 17\} \), which is also false because we have \( 26 \) in the range.

Thus, the false statement is **Option 3**.