The domain of R is :
`R={x,y):x+2y=8 , x,y in N}`

Options

A. {2,4,8}
B. {2,4,6,8}
C. {2,4,6}
D. {1,2,3,4}

Answer

To find the domain of the relation \( R = \{(x, y) : x + 2y = 8, \, x, y \in \mathbb{N}\} \), we will follow these steps:

### Step 1: Understand the equation
We start with the equation given in the relation:
\[ x + 2y = 8 \]
where \( x \) and \( y \) are natural numbers.

### Step 2: Rearrange the equation
We can rearrange the equation to express \( x \) in terms of \( y \):
\[ x = 8 - 2y \]

### Step 3: Determine the values of \( y \)
Since \( x \) and \( y \) must be natural numbers (i.e., \( \mathbb{N} = \{1, 2, 3, \ldots\} \)), we need to find suitable values for \( y \) such that \( x \) remains a natural number.

### Step 4: Substitute values for \( y \)
We will substitute natural number values for \( y \) and calculate the corresponding \( x \):

- For \( y = 1 \):
\[ x = 8 - 2(1) = 8 - 2 = 6 \quad \Rightarrow \quad (6, 1) \]

- For \( y = 2 \):
\[ x = 8 - 2(2) = 8 - 4 = 4 \quad \Rightarrow \quad (4, 2) \]

- For \( y = 3 \):
\[ x = 8 - 2(3) = 8 - 6 = 2 \quad \Rightarrow \quad (2, 3) \]

- For \( y = 4 \):
\[ x = 8 - 2(4) = 8 - 8 = 0 \quad \Rightarrow \quad (0, 4) \quad \text{(not valid since 0 is not in } \mathbb{N\text{)} \]

### Step 5: List valid pairs
The valid pairs we found are:
1. \( (6, 1) \)
2. \( (4, 2) \)
3. \( (2, 3) \)

### Step 6: Identify the domain
The domain of the relation \( R \) consists of the first elements of the valid pairs:
\[ \text{Domain } R = \{6, 4, 2\} \]

### Conclusion
Thus, the domain of \( R \) is:
\[ \{2, 4, 6\} \]