Let R be the relation in the set N, given by
`R={(x,y):x=y+3,ygt5}`. Choose the correct answer from the following:

Options

A. `(7,4)inR`
B. `(9,6)inR`
C. `(4,7)inR`
D. `(8,5)inR`

Answer

To solve the problem, we need to analyze the relation \( R \) defined by \( R = \{(x,y) : x = y + 3, y > 5\} \) and determine which option satisfies this relation.

### Step-by-Step Solution:

1. **Understand the Relation**: The relation states that for any pair \( (x, y) \) in \( R \), \( x \) must equal \( y + 3 \) and \( y \) must be greater than 5.

2. **Identify Possible Values for \( y \)**: Since \( y \) must be greater than 5, the smallest integer value for \( y \) is 6. Therefore, we can start checking values of \( y \) from 6 onwards.

3. **Calculate Corresponding \( x \) Values**:
- For \( y = 6 \):
\[ x = y + 3 = 6 + 3 = 9 \]
So, the pair \( (9, 6) \) is in the relation \( R \).
- For \( y = 7 \):
\[ x = y + 3 = 7 + 3 = 10 \]
So, the pair \( (10, 7) \) is in the relation \( R \).
- For \( y = 8 \):
\[ x = y + 3 = 8 + 3 = 11 \]
So, the pair \( (11, 8) \) is in the relation \( R \).
- And so on...

4. **Check the Options**: Now we check the provided options to see which one fits the relation \( R \):
- **Option A**: \( y < 5 \) - This is incorrect since \( y \) must be greater than 5.
- **Option B**: \( x = 9, y = 6 \) - Here, \( y = 6 \) is greater than 5, and \( x = 9 \) satisfies \( x = y + 3 \). This option is correct.
- **Option C**: \( x = 4, y = 7 \) - Here, \( y = 7 \) is greater than 5, but \( x \) does not satisfy \( x = y + 3 \) since \( 4 \neq 7 + 3 \). This option is incorrect.
- **Option D**: \( y = 5 \) - This is incorrect since \( y \) must be greater than 5.

5. **Conclusion**: The only option that satisfies the relation \( R \) is **Option B**: \( (9, 6) \).