If `R={(x,y):x+2y=10}` is a relation in N, write the range of R.

Answer

To find the range of the relation \( R = \{(x, y) : x + 2y = 10\} \) where \( x \) and \( y \) are natural numbers, we will follow these steps:

### Step-by-Step Solution:

1. **Understand the equation**: We start with the equation given by the relation, which is \( x + 2y = 10 \).

2. **Rearrange the equation**: We can express \( y \) in terms of \( x \):
\[ 2y = 10 - x \implies y = \frac{10 - x}{2} \]

3. **Identify the domain**: Since \( x \) and \( y \) must be natural numbers (positive integers), we need to find valid values for \( x \) such that \( y \) remains a natural number.

4. **Determine values for \( x \)**:
- For \( y \) to be a natural number, \( 10 - x \) must be even (since it is divided by 2). This means \( x \) must be even.
- The possible even values for \( x \) that are less than or equal to 10 are \( 2, 4, 6, 8, 10 \).

5. **Calculate corresponding \( y \) values**:
- If \( x = 2 \):
\[ y = \frac{10 - 2}{2} = \frac{8}{2} = 4 \]
- If \( x = 4 \):
\[ y = \frac{10 - 4}{2} = \frac{6}{2} = 3 \]
- If \( x = 6 \):
\[ y = \frac{10 - 6}{2} = \frac{4}{2} = 2 \]
- If \( x = 8 \):
\[ y = \frac{10 - 8}{2} = \frac{2}{2} = 1 \]
- If \( x = 10 \):
\[ y = \frac{10 - 10}{2} = \frac{0}{2} = 0 \quad (\text{not a natural number}) \]

6. **List the valid pairs**: The valid pairs \((x, y)\) are:
- \( (2, 4) \)
- \( (4, 3) \)
- \( (6, 2) \)
- \( (8, 1) \)

7. **Extract the range**: From the valid pairs, the corresponding \( y \) values are \( 4, 3, 2, 1 \).

8. **Write the final range**: Therefore, the range of \( R \) is:
\[ \text{Range} = \{1, 2, 3, 4\} \]

### Final Answer:
The range of the relation \( R \) is \( \{1, 2, 3, 4\} \).

---