Let R be a relation in N defined by `R={(x, y):2x+y=8}`, then range of R is

Options

A. `{1, 2, 3}`
B. `{2, 4, 6}`
C. `{1, 2, 3, 4, 6}`
D. none of these

Answer

To find the range of the relation \( R = \{(x, y) : 2x + y = 8\} \) where \( x \) and \( y \) are natural numbers, we can follow these steps:

### Step 1: Express \( y \) in terms of \( x \)
From the equation \( 2x + y = 8 \), we can isolate \( y \):
\[ y = 8 - 2x \]

### Step 2: Determine the possible values of \( x \)
Since \( x \) must be a natural number, we start with the smallest natural number, which is 1. We will find the corresponding values of \( y \) for different values of \( x \).

### Step 3: Calculate \( y \) for different values of \( x \)
- If \( x = 1 \):
\[ y = 8 - 2(1) = 8 - 2 = 6 \quad \text{(valid, since 6 is a natural number)} \]
- If \( x = 2 \):
\[ y = 8 - 2(2) = 8 - 4 = 4 \quad \text{(valid, since 4 is a natural number)} \]
- If \( x = 3 \):
\[ y = 8 - 2(3) = 8 - 6 = 2 \quad \text{(valid, since 2 is a natural number)} \]
- If \( x = 4 \):
\[ y = 8 - 2(4) = 8 - 8 = 0 \quad \text{(not valid, since 0 is not a natural number)} \]

### Step 4: List the valid pairs \((x, y)\)
From the calculations, we have the following valid pairs:
- \( (1, 6) \)
- \( (2, 4) \)
- \( (3, 2) \)

### Step 5: Identify the range of \( R \)
The range of the relation \( R \) consists of the second elements \( y \) of the valid pairs:
- From \( (1, 6) \), we get \( 6 \)
- From \( (2, 4) \), we get \( 4 \)
- From \( (3, 2) \), we get \( 2 \)

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

### Step 6: Arrange the range in increasing order
The final range in increasing order is:
\[ \{2, 4, 6\} \]

### Final Answer
The range of the relation \( R \) is \( \{2, 4, 6\} \).

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