A relation R is defined from the set of integer Z to Z as follows:
`(x,y) in Z hArr x^(2) + y^(2)=25`
(i) Express R and `R^(-1)` as the set of ordered pairs.
(ii) Write the domain of R and `R^(-1)`.

Answer

To solve the problem step-by-step, we need to express the relation \( R \) and its inverse \( R^{-1} \) as sets of ordered pairs, and then determine the domains of both relations.

### Step 1: Find the ordered pairs for relation \( R \)

We start with the equation defining the relation:
\[ x^2 + y^2 = 25 \]
We will find integer pairs \((x, y)\) that satisfy this equation.

1. **For \( x = 0 \)**:
\[ y^2 = 25 \implies y = 5 \text{ or } y = -5 \]
Ordered pairs: \( (0, 5), (0, -5) \)

2. **For \( x = 1 \)**:
\[ y^2 = 25 - 1^2 = 24 \text{ (not a perfect square)} \]
No valid pairs.

3. **For \( x = 2 \)**:
\[ y^2 = 25 - 2^2 = 21 \text{ (not a perfect square)} \]
No valid pairs.

4. **For \( x = 3 \)**:
\[ y^2 = 25 - 3^2 = 16 \implies y = 4 \text{ or } y = -4 \]
Ordered pairs: \( (3, 4), (3, -4) \)

5. **For \( x = 4 \)**:
\[ y^2 = 25 - 4^2 = 9 \implies y = 3 \text{ or } y = -3 \]
Ordered pairs: \( (4, 3), (4, -3) \)

6. **For \( x = 5 \)**:
\[ y^2 = 25 - 5^2 = 0 \implies y = 0 \]
Ordered pairs: \( (5, 0) \)

7. **For \( x = -1, -2, -3, -4, -5 \)**:
By symmetry, the pairs will be the same as for positive \( x \):
- For \( x = -3 \): \( (-3, 4), (-3, -4) \)
- For \( x = -4 \): \( (-4, 3), (-4, -3) \)
- For \( x = -5 \): \( (-5, 0) \)

Now, we can compile all the ordered pairs:
\[ R = \{ (0, 5), (0, -5), (3, 4), (3, -4), (4, 3), (4, -3), (5, 0), (-3, 4), (-3, -4), (-4, 3), (-4, -3), (-5, 0) \} \]

### Step 2: Find the ordered pairs for relation \( R^{-1} \)

The inverse relation \( R^{-1} \) is obtained by swapping the elements of each ordered pair in \( R \):
\[ R^{-1} = \{ (5, 0), (-5, 0), (4, 3), (-4, 3), (4, -3), (-4, -3), (0, 5), (0, -5), (3, 4), (3, -4), (-3, 4), (-3, -4) \} \]

### Step 3: Determine the domains of \( R \) and \( R^{-1} \)

1. **Domain of \( R \)**:
The domain consists of the first elements (x-values) of the ordered pairs in \( R \):
\[ \text{Domain of } R = \{ 0, 3, 4, 5, -3, -4, -5 \} \]

2. **Domain of \( R^{-1} \)**:
The domain consists of the first elements (x-values) of the ordered pairs in \( R^{-1} \):
\[ \text{Domain of } R^{-1} = \{ 5, -5, 4, -4, 0, 3, -3 \} \]

### Final Answers

- **Relation \( R \)**:
\[ R = \{ (0, 5), (0, -5), (3, 4), (3, -4), (4, 3), (4, -3), (5, 0), (-3, 4), (-3, -4), (-4, 3), (-4, -3), (-5, 0) \} \]

- **Inverse Relation \( R^{-1} \)**:
\[ R^{-1} = \{ (5, 0), (-5, 0), (4, 3), (-4, 3), (4, -3), (-4, -3), (0, 5), (0, -5), (3, 4), (3, -4), (-3, 4), (-3, -4) \} \]

- **Domain of \( R \)**:
\[ \{ 0, 3, 4, 5, -3, -4, -5 \} \]

- **Domain of \( R^{-1} \)**:
\[ \{ 5, -5, 4, -4, 0, 3, -3 \} \]