Let R be the relation in the set N given by `R={(a,b):a=b-2, b gt 6}`, then:

Options

A. `(2,4) in R`
B. `(3,8) in R`
C. `(6,8) in R`
D. `(8,7) in R`

Answer

To solve the problem, we need to analyze the relation \( R \) defined as \( R = \{(a, b) : a = b - 2, b > 6\} \) where \( a \) and \( b \) are natural numbers.

### Step-by-Step Solution:

1. **Understanding the Relation**:
The relation states that for every pair \( (a, b) \), \( a \) is equal to \( b - 2 \) and \( b \) must be greater than 6. This means we can express \( b \) in terms of \( a \):
\[ b = a + 2 \]

2. **Finding the Minimum Value of \( b \)**:
Since \( b > 6 \), we can set up the inequality:
\[ a + 2 > 6 \]
Subtracting 2 from both sides gives:
\[ a > 4 \]
This means \( a \) must be greater than 4.

3. **Finding Possible Values of \( a \)**:
Since \( a \) is a natural number, the smallest possible value for \( a \) is 5. We can now find corresponding values of \( b \):
- If \( a = 5 \):
\[ b = 5 + 2 = 7 \quad \Rightarrow \quad (5, 7) \]
- If \( a = 6 \):
\[ b = 6 + 2 = 8 \quad \Rightarrow \quad (6, 8) \]
- If \( a = 7 \):
\[ b = 7 + 2 = 9 \quad \Rightarrow \quad (7, 9) \]
- Continuing this way, we can see that for any natural number \( a \geq 5 \), we can find a corresponding \( b \).

4. **Listing the Relation**:
Thus, the relation \( R \) can be expressed as:
\[ R = \{(5, 7), (6, 8), (7, 9), (8, 10), \ldots\} \]
In general, the pairs will be of the form \( (a, a + 2) \) for \( a \geq 5 \).

5. **Conclusion**:
The relation \( R \) consists of all ordered pairs \( (a, b) \) where \( a \) is a natural number greater than or equal to 5, and \( b \) is \( a + 2 \).