If `R ={(x,y): x,y in Z , x^2 +y^2 =64}` then , write R in the roaster form

Answer

To express the relation \( R = \{(x,y) : x,y \in \mathbb{Z}, x^2 + y^2 = 64\} \) in roster form, we need to find all integer pairs \((x, y)\) such that the sum of their squares equals 64.

### Step-by-Step Solution:

1. **Identify the maximum values for \(x\) and \(y\)**:
Since \(x^2 + y^2 = 64\), both \(x\) and \(y\) must be integers whose squares do not exceed 64. The maximum integer \(x\) can take is 8 (since \(8^2 = 64\)) and the minimum is -8.

2. **Calculate possible values for \(y\)**:
For each integer value of \(x\) from -8 to 8, we will calculate \(y\) using the equation \(y^2 = 64 - x^2\).

3. **Evaluate for each integer \(x\)**:
- For \(x = 0\):
\[ y^2 = 64 - 0^2 = 64 \implies y = \pm 8 \quad \Rightarrow \quad (0, 8), (0, -8) \]
- For \(x = 1\):
\[ y^2 = 64 - 1^2 = 63 \quad \text{(not a perfect square)} \]
- For \(x = 2\):
\[ y^2 = 64 - 2^2 = 60 \quad \text{(not a perfect square)} \]
- For \(x = 3\):
\[ y^2 = 64 - 3^2 = 55 \quad \text{(not a perfect square)} \]
- For \(x = 4\):
\[ y^2 = 64 - 4^2 = 48 \quad \text{(not a perfect square)} \]
- For \(x = 5\):
\[ y^2 = 64 - 5^2 = 39 \quad \text{(not a perfect square)} \]
- For \(x = 6\):
\[ y^2 = 64 - 6^2 = 28 \quad \text{(not a perfect square)} \]
- For \(x = 7\):
\[ y^2 = 64 - 7^2 = 15 \quad \text{(not a perfect square)} \]
- For \(x = 8\):
\[ y^2 = 64 - 8^2 = 0 \implies y = 0 \quad \Rightarrow \quad (8, 0) \]

4. **Consider negative values for \(x\)**:
The relation is symmetric, so we can use the results from positive \(x\) values:
- For \(x = -1\) to \(x = -7\), the results will be the same as for \(x = 1\) to \(x = 7\) (not perfect squares).
- For \(x = -8\):
\[ y^2 = 64 - (-8)^2 = 0 \implies y = 0 \quad \Rightarrow \quad (-8, 0) \]

5. **Compile the results**:
The integer pairs \((x, y)\) that satisfy \(x^2 + y^2 = 64\) are:
- \((0, 8)\)
- \((0, -8)\)
- \((8, 0)\)
- \((-8, 0)\)

Thus, the relation \( R \) in roster form is:
\[ R = \{(0, 8), (0, -8), (8, 0), (-8, 0)\} \]