If A = {2,3,4,5,6) and R is a relation on set A defined by `R= {(x, y): y=x+2, X, y in A}` then

Options

A. {(2,4), (3,5), (4, 6)}
B. {(4,2), (5,3), (6,4)}
C. {(2,4), (5,3), (4, 6)}
D. {(4,2), (3,5),(4,6)}

Answer

To solve the problem, we need to determine the relation \( R \) defined on the set \( A = \{2, 3, 4, 5, 6\} \) such that \( R = \{(x, y) : y = x + 2, x, y \in A\} \).

### Step-by-Step Solution:

1. **Identify the Set A**:
We have the set \( A = \{2, 3, 4, 5, 6\} \).

2. **Define the Relation**:
The relation \( R \) is defined such that for each \( x \) in \( A \), \( y \) must equal \( x + 2 \).

3. **Calculate Possible Pairs**:
We will check each element \( x \) in set \( A \) to see if \( y = x + 2 \) also belongs to set \( A \):
- For \( x = 2 \):
\[ y = 2 + 2 = 4 \quad (\text{4 is in } A) \]
So, the pair \( (2, 4) \) is included in \( R \).
- For \( x = 3 \):
\[ y = 3 + 2 = 5 \quad (\text{5 is in } A) \]
So, the pair \( (3, 5) \) is included in \( R \).
- For \( x = 4 \):
\[ y = 4 + 2 = 6 \quad (\text{6 is in } A) \]
So, the pair \( (4, 6) \) is included in \( R \).
- For \( x = 5 \):
\[ y = 5 + 2 = 7 \quad (\text{7 is not in } A) \]
So, the pair \( (5, 7) \) is not included in \( R \).
- For \( x = 6 \):
\[ y = 6 + 2 = 8 \quad (\text{8 is not in } A) \]
So, the pair \( (6, 8) \) is not included in \( R \).

4. **Compile the Relation R**:
Based on the calculations, the pairs that satisfy the condition \( y = x + 2 \) and where both \( x \) and \( y \) belong to set \( A \) are:
\[ R = \{(2, 4), (3, 5), (4, 6)\} \]

### Final Answer:
Thus, the relation \( R \) is:
\[ R = \{(2, 4), (3, 5), (4, 6)\} \]