A is a set of first 10 natural numbers and R is a relation from A to A defined as:
`(x,y) in R hArr x+2y=10` when `x,y in A`
(i)Express R in the form of a set of ordered pairs.
(ii) Find the domain and range of R.
(iii) Find `R^(-1)` .

Answer

To solve the problem step by step, we will follow the instructions given in the question.

### Step 1: Express R in the form of a set of ordered pairs

Given that \( A \) is the set of the first 10 natural numbers:
\[ A = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \]

The relation \( R \) is defined as:
\[ (x, y) \in R \iff x + 2y = 10 \]
where \( x, y \in A \).

To find the ordered pairs in \( R \), we will substitute values of \( x \) from the set \( A \) and solve for \( y \).

1. **For \( x = 1 \)**:
\[ 1 + 2y = 10 \implies 2y = 9 \implies y = \frac{9}{2} \quad (\text{not a natural number}) \]

2. **For \( x = 2 \)**:
\[ 2 + 2y = 10 \implies 2y = 8 \implies y = 4 \quad (\text{valid}) \]
Ordered pair: \( (2, 4) \)

3. **For \( x = 3 \)**:
\[ 3 + 2y = 10 \implies 2y = 7 \implies y = \frac{7}{2} \quad (\text{not a natural number}) \]

4. **For \( x = 4 \)**:
\[ 4 + 2y = 10 \implies 2y = 6 \implies y = 3 \quad (\text{valid}) \]
Ordered pair: \( (4, 3) \)

5. **For \( x = 5 \)**:
\[ 5 + 2y = 10 \implies 2y = 5 \implies y = \frac{5}{2} \quad (\text{not a natural number}) \]

6. **For \( x = 6 \)**:
\[ 6 + 2y = 10 \implies 2y = 4 \implies y = 2 \quad (\text{valid}) \]
Ordered pair: \( (6, 2) \)

7. **For \( x = 7 \)**:
\[ 7 + 2y = 10 \implies 2y = 3 \implies y = \frac{3}{2} \quad (\text{not a natural number}) \]

8. **For \( x = 8 \)**:
\[ 8 + 2y = 10 \implies 2y = 2 \implies y = 1 \quad (\text{valid}) \]
Ordered pair: \( (8, 1) \)

9. **For \( x = 9 \)**:
\[ 9 + 2y = 10 \implies 2y = 1 \implies y = \frac{1}{2} \quad (\text{not a natural number}) \]

10. **For \( x = 10 \)**:
\[ 10 + 2y = 10 \implies 2y = 0 \implies y = 0 \quad (\text{not a natural number}) \]

Thus, the relation \( R \) in the form of a set of ordered pairs is:
\[ R = \{(2, 4), (4, 3), (6, 2), (8, 1)\} \]

### Step 2: Find the domain and range of R

- **Domain**: The set of all first elements (x-values) in the ordered pairs of \( R \):
\[ \text{Domain} = \{2, 4, 6, 8\} \]

- **Range**: The set of all second elements (y-values) in the ordered pairs of \( R \):
\[ \text{Range} = \{4, 3, 2, 1\} \]

### Step 3: Find \( R^{-1} \)

To find the inverse relation \( R^{-1} \), we swap the elements of each ordered pair in \( R \):
\[ R^{-1} = \{(4, 2), (3, 4), (2, 6), (1, 8)\} \]

### Summary of Results

1. \( R = \{(2, 4), (4, 3), (6, 2), (8, 1)\} \)
2. Domain of \( R = \{2, 4, 6, 8\} \)
3. Range of \( R = \{4, 3, 2, 1\} \)
4. \( R^{-1} = \{(4, 2), (3, 4), (2, 6), (1, 8)\} \)