The relation R defined on the set of natural numbers as {(a,b) : a differs from b by 3} is given by

Options

A. `{(1,4),(2,5),(3,6),...}`
B. `{(4,1),(5,2),(6,3),...}`
C. `{(1,3),(2,6),(3,9),...}`
D. None of these

Answer

To solve the problem, we need to define the relation \( R \) on the set of natural numbers \( \mathbb{N} \) such that the ordered pair \( (a, b) \) satisfies the condition that \( a \) differs from \( b \) by 3. This means we can express this relationship mathematically.

### Step-by-Step Solution:

1. **Understanding the Relation**:
The relation \( R \) is defined as \( R = \{(a, b) : a - b = 3\} \). This means that for any two natural numbers \( a \) and \( b \), the difference between \( a \) and \( b \) should be exactly 3.

2. **Rearranging the Equation**:
From the equation \( a - b = 3 \), we can rearrange it to express \( a \) in terms of \( b \):
\[ a = b + 3 \]

3. **Considering Natural Numbers**:
Since \( a \) and \( b \) are both natural numbers, we need to ensure that \( b \) is a natural number such that \( a \) also remains a natural number. Therefore, \( b \) must be at least 1 (the smallest natural number).

4. **Generating Pairs**:
Now, we can generate pairs \( (a, b) \) based on the values of \( b \):
- If \( b = 1 \), then \( a = 1 + 3 = 4 \) → Pair: \( (4, 1) \)
- If \( b = 2 \), then \( a = 2 + 3 = 5 \) → Pair: \( (5, 2) \)
- If \( b = 3 \), then \( a = 3 + 3 = 6 \) → Pair: \( (6, 3) \)
- If \( b = 4 \), then \( a = 4 + 3 = 7 \) → Pair: \( (7, 4) \)
- Continuing this way, we can see that for any natural number \( b \), \( a \) will always be \( b + 3 \).

5. **Writing the Relation**:
Therefore, the relation \( R \) can be expressed as:
\[ R = \{(b + 3, b) : b \in \mathbb{N}\} \]
This means that for every natural number \( b \), there exists a corresponding \( a \) which is \( b + 3 \).

6. **Identifying the Correct Option**:
Based on the generated pairs, we can check which of the provided options matches our relation. The pairs we generated are:
- \( (4, 1) \)
- \( (5, 2) \)
- \( (6, 3) \)
- \( (7, 4) \)
- and so on...

We compare these pairs with the options given in the question to find the correct answer.

### Conclusion:
After checking the options, we find that the correct option that matches our derived relation is option 2.