If R = `{(x,y): x, y in W, x^(2) + y^(2) = 25}` , then find the domain and range of R .

Answer

To find the domain and range of the relation \( R = \{(x,y): x, y \in W, x^2 + y^2 = 25\} \), we will follow these steps:

### Step 1: Understand the equation
The equation \( x^2 + y^2 = 25 \) represents a circle with a radius of 5 centered at the origin (0,0) in the Cartesian plane. However, since \( x \) and \( y \) must be whole numbers (elements of \( W \), the set of whole numbers), we need to find the integer solutions to this equation.

### Step 2: Find integer solutions
We will find pairs \( (x, y) \) such that both \( x \) and \( y \) are whole numbers and satisfy the equation \( x^2 + y^2 = 25 \).

1. **When \( x = 0 \)**:
\[ 0^2 + y^2 = 25 \implies y^2 = 25 \implies y = 5 \]
So, we have the point \( (0, 5) \).

2. **When \( x = 3 \)**:
\[ 3^2 + y^2 = 25 \implies 9 + y^2 = 25 \implies y^2 = 16 \implies y = 4 \]
So, we have the point \( (3, 4) \).

3. **When \( x = 4 \)**:
\[ 4^2 + y^2 = 25 \implies 16 + y^2 = 25 \implies y^2 = 9 \implies y = 3 \]
So, we have the point \( (4, 3) \).

4. **When \( x = 5 \)**:
\[ 5^2 + y^2 = 25 \implies 25 + y^2 = 25 \implies y^2 = 0 \implies y = 0 \]
So, we have the point \( (5, 0) \).

### Step 3: List all pairs
The integer pairs that satisfy the equation \( x^2 + y^2 = 25 \) are:
- \( (0, 5) \)
- \( (3, 4) \)
- \( (4, 3) \)
- \( (5, 0) \)

### Step 4: Determine the domain
The domain of \( R \) consists of the first elements (x-coordinates) of the ordered pairs:
- From the pairs, the x-coordinates are \( 0, 3, 4, 5 \).
- Therefore, the domain is:
\[ \text{Domain} = \{0, 3, 4, 5\} \]

### Step 5: Determine the range
The range of \( R \) consists of the second elements (y-coordinates) of the ordered pairs:
- From the pairs, the y-coordinates are \( 5, 4, 3, 0 \).
- Therefore, the range is:
\[ \text{Range} = \{5, 4, 3, 0\} \]

### Final Answer
Thus, the domain and range of the relation \( R \) are:
- **Domain**: \( \{0, 3, 4, 5\} \)
- **Range**: \( \{5, 4, 3, 0\} \)