A relation R is defined in the set Z of integers as follows (x, y ) `in` R iff `x^(2) + y^(2)` = 9 . Which of the following is false ?

Options

A. R = { (0,3), (0,-3), (3,0), (-3,0)}
B. Domain of R = { - 3, 0, 3}
C. Range of R = {-3, 0, 3}
D. None of these

Answer

To solve the problem, we need to analyze the relation \( R \) defined on the set of integers \( \mathbb{Z} \) such that \( (x, y) \in R \) if and only if \( x^2 + y^2 = 9 \). We will find the ordered pairs that satisfy this equation and then determine the domain and range of the relation.

### Step-by-Step Solution:

1. **Understanding the Equation**:
We start with the equation \( x^2 + y^2 = 9 \). This represents a circle with a radius of 3 centered at the origin in the coordinate system.

2. **Finding Integer Solutions**:
We need to find all integer pairs \( (x, y) \) that satisfy this equation. We can do this by considering possible values for \( x \) and calculating \( y \).

- If \( x = 0 \):
\[ 0^2 + y^2 = 9 \implies y^2 = 9 \implies y = 3 \text{ or } y = -3 \implies (0, 3), (0, -3) \]

- If \( x = 1 \):
\[ 1^2 + y^2 = 9 \implies 1 + y^2 = 9 \implies y^2 = 8 \text{ (not an integer)} \]

- If \( x = 2 \):
\[ 2^2 + y^2 = 9 \implies 4 + y^2 = 9 \implies y^2 = 5 \text{ (not an integer)} \]

- If \( x = 3 \):
\[ 3^2 + y^2 = 9 \implies 9 + y^2 = 9 \implies y^2 = 0 \implies y = 0 \implies (3, 0) \]

- If \( x = -1 \):
\[ (-1)^2 + y^2 = 9 \implies 1 + y^2 = 9 \implies y^2 = 8 \text{ (not an integer)} \]

- If \( x = -2 \):
\[ (-2)^2 + y^2 = 9 \implies 4 + y^2 = 9 \implies y^2 = 5 \text{ (not an integer)} \]

- If \( x = -3 \):
\[ (-3)^2 + y^2 = 9 \implies 9 + y^2 = 9 \implies y^2 = 0 \implies y = 0 \implies (-3, 0) \]

3. **Compiling the Ordered Pairs**:
From the calculations above, the integer pairs that satisfy \( x^2 + y^2 = 9 \) are:
- \( (0, 3) \)
- \( (0, -3) \)
- \( (3, 0) \)
- \( (-3, 0) \)

Thus, the relation \( R \) can be expressed as:
\[ R = \{ (0, 3), (0, -3), (3, 0), (-3, 0) \} \]

4. **Finding the Domain and Range**:
- **Domain**: The set of all first elements (x-coordinates) in the ordered pairs:
\[ \text{Domain} = \{ 0, 3, -3 \} \]

- **Range**: The set of all second elements (y-coordinates) in the ordered pairs:
\[ \text{Range} = \{ 3, -3, 0 \} \]

5. **Identifying the False Statement**:
We need to determine which of the provided options is false. Since the domain and range have been correctly identified, we can conclude that options A, B, and C are true, and thus the correct answer is option D, which states that none of the statements is false.

### Conclusion:
The final answer is option D: None of these statements is false.