If `R={(x,y):x,y in W,2x+y=8}`, then domain of R is

Options

A. {0, 1, 2, 3, 4, 5}
B. {0,1,2,3,4,5,6}
C. {0,1,2,3,4}
D. {0, 1, 2, 3}

Answer

To find the domain of the relation \( R = \{(x, y) : x, y \in W, 2x + y = 8\} \), where \( W \) represents the set of whole numbers, we will follow these steps:

### Step 1: Understand the equation
The equation \( 2x + y = 8 \) can be rearranged to express \( y \) in terms of \( x \):
\[ y = 8 - 2x \]

### Step 2: Determine the values of \( x \)
Since \( x \) must be a whole number (i.e., \( x \in W \)), we will substitute whole number values for \( x \) and check if \( y \) remains a whole number.

### Step 3: Substitute values for \( x \)
1. **For \( x = 0 \)**:
\[ y = 8 - 2(0) = 8 \quad \text{(valid, as } y \in W\text{)} \]
2. **For \( x = 1 \)**:
\[ y = 8 - 2(1) = 6 \quad \text{(valid)} \]
3. **For \( x = 2 \)**:
\[ y = 8 - 2(2) = 4 \quad \text{(valid)} \]
4. **For \( x = 3 \)**:
\[ y = 8 - 2(3) = 2 \quad \text{(valid)} \]
5. **For \( x = 4 \)**:
\[ y = 8 - 2(4) = 0 \quad \text{(valid)} \]
6. **For \( x = 5 \)**:
\[ y = 8 - 2(5) = -2 \quad \text{(invalid, as } y \notin W\text{)} \]

### Step 4: Identify the maximum value of \( x \)
Since \( y \) must be a whole number, we observe that for \( x \geq 5 \), \( y \) becomes negative. Thus, the largest value of \( x \) that keeps \( y \) in the set of whole numbers is \( x = 4 \).

### Step 5: List the domain
The valid whole number values for \( x \) are:
\[ x = 0, 1, 2, 3, 4 \]
Thus, the domain of \( R \) is:
\[ \text{Domain of } R = \{0, 1, 2, 3, 4\} \]

### Final Answer:
The domain of \( R \) is \( \{0, 1, 2, 3, 4\} \).

---