The domain of relation
R = {(x,y) : `x^(2) + y^(2) = 16, x, y in` Z } is

Options

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

Answer

To find the domain of the relation \( R = \{(x,y) : x^2 + y^2 = 16, x, y \in \mathbb{Z}\} \), we need to determine the integer pairs \((x, y)\) that satisfy the equation \( x^2 + y^2 = 16 \).

### Step-by-Step Solution:

1. **Understand the Equation**: The equation \( x^2 + y^2 = 16 \) represents a circle with a radius of 4 centered at the origin (0,0) in the Cartesian plane.

2. **Identify Possible Integer Values**: Since \( x \) and \( y \) must be integers, we will find integer solutions for \( x^2 + y^2 = 16 \).

3. **Calculate Possible Values for \( x \)**:
- The maximum value for \( x \) occurs when \( y = 0 \):
\[ x^2 = 16 \implies x = 4 \text{ or } x = -4 \]
- Therefore, \( x \) can take values in the range from \(-4\) to \(4\).

4. **Find Corresponding \( y \) Values**:
- For each integer value of \( x \) from \(-4\) to \(4\), we calculate \( y \) using \( y^2 = 16 - x^2 \):
- If \( x = 0 \): \( y^2 = 16 \implies y = 4 \text{ or } -4 \) → Pairs: (0, 4), (0, -4)
- If \( x = 4 \): \( y^2 = 0 \implies y = 0 \) → Pair: (4, 0)
- If \( x = -4 \): \( y^2 = 0 \implies y = 0 \) → Pair: (-4, 0)
- If \( x = 3 \): \( y^2 = 7 \) → No integer solutions.
- If \( x = -3 \): \( y^2 = 7 \) → No integer solutions.
- If \( x = 2 \): \( y^2 = 12 \) → No integer solutions.
- If \( x = -2 \): \( y^2 = 12 \) → No integer solutions.
- If \( x = 1 \): \( y^2 = 15 \) → No integer solutions.
- If \( x = -1 \): \( y^2 = 15 \) → No integer solutions.

5. **List All Ordered Pairs**:
- The integer solutions to \( x^2 + y^2 = 16 \) are:
- (0, 4)
- (0, -4)
- (4, 0)
- (-4, 0)

6. **Determine the Domain**:
- The domain of the relation \( R \) consists of the first elements of the ordered pairs found:
- From the pairs: (0, 4), (0, -4), (4, 0), (-4, 0)
- The unique values of \( x \) are \( 0, 4, -4 \).

7. **Final Answer**:
- Therefore, the domain of the relation \( R \) is:
\[ \{0, 4, -4\} \]