If A={1,2,3,4} and B={5,7,8,11,15}, are two sets and a relation R from A to B is defined as follows:
`""_(x)R_(y) hArr y= 2x+3`, where `x in A, y in B`
(i) Express R in Roaster form.
(ii) Find the domain and range of R.
(iii) Find `R^(-1)` .
(iv) Represent R by arrow diagram.

Answer

Let's solve the problem step by step.

### Given:
- Set A = {1, 2, 3, 4}
- Set B = {5, 7, 8, 11, 15}
- Relation R is defined as \( y = 2x + 3 \) where \( x \in A \) and \( y \in B \).

### (i) Express R in Roaster form.

To express R in roster form, we need to find pairs \( (x, y) \) such that \( y = 2x + 3 \) and \( y \) belongs to set B.

1. For \( x = 1 \):
\[ y = 2(1) + 3 = 2 + 3 = 5 \quad (\text{5 is in B}) \]
Pair: \( (1, 5) \)

2. For \( x = 2 \):
\[ y = 2(2) + 3 = 4 + 3 = 7 \quad (\text{7 is in B}) \]
Pair: \( (2, 7) \)

3. For \( x = 3 \):
\[ y = 2(3) + 3 = 6 + 3 = 9 \quad (\text{9 is NOT in B}) \]
No pair for \( x = 3 \).

4. For \( x = 4 \):
\[ y = 2(4) + 3 = 8 + 3 = 11 \quad (\text{11 is in B}) \]
Pair: \( (4, 11) \)

Thus, the relation R in roster form is:
\[ R = \{(1, 5), (2, 7), (4, 11)\} \]

### (ii) Find the domain and range of R.

- **Domain**: The set of all first elements (x-values) in the pairs of R.
- From R: \( \{1, 2, 4\} \)

- **Range**: The set of all second elements (y-values) in the pairs of R.
- From R: \( \{5, 7, 11\} \)

Thus,
- Domain = \( \{1, 2, 4\} \)
- Range = \( \{5, 7, 11\} \)

### (iii) Find \( R^{-1} \).

To find the inverse relation \( R^{-1} \), we reverse each pair in R.

From \( R = \{(1, 5), (2, 7), (4, 11)\} \):
- Reverse \( (1, 5) \) to \( (5, 1) \)
- Reverse \( (2, 7) \) to \( (7, 2) \)
- Reverse \( (4, 11) \) to \( (11, 4) \)

Thus, the inverse relation \( R^{-1} \) is:
\[ R^{-1} = \{(5, 1), (7, 2), (11, 4)\} \]

### (iv) Represent R by arrow diagram.

To represent R using an arrow diagram:
1. Draw two circles, one for set A and one for set B.
2. Write the elements of set A (1, 2, 3, 4) in the left circle and the elements of set B (5, 7, 8, 11, 15) in the right circle.
3. Draw arrows from elements of A to their corresponding elements in B based on the pairs in R:
- Draw an arrow from 1 to 5.
- Draw an arrow from 2 to 7.
- Draw an arrow from 4 to 11.
4. No arrow for 3 since there is no corresponding y in B.

### Final Answers:
- (i) \( R = \{(1, 5), (2, 7), (4, 11)\} \)
- (ii) Domain = \( \{1, 2, 4\} \), Range = \( \{5, 7, 11\} \)
- (iii) \( R^{-1} = \{(5, 1), (7, 2), (11, 4)\} \)