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 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

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

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

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

1. **Calculate the values of y for each x in A:**
- For \( x = 1 \): \( y = 2(1) + 3 = 5 \)
- For \( x = 2 \): \( y = 2(2) + 3 = 7 \)
- For \( x = 3 \): \( y = 2(3) + 3 = 9 \)
- For \( x = 4 \): \( y = 2(4) + 3 = 11 \)

2. **Identify the valid pairs (x, y) where y is in set B:**
- From the calculations:
- \( (1, 5) \)
- \( (2, 7) \)
- \( (3, 9) \) (not in B)
- \( (4, 11) \)

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

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

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

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

3. **Thus, the domain and range are:**
- Domain: \( \{1, 2, 4\} \)
- Range: \( \{5, 7, 11\} \)

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

1. **To find the inverse relation \( R^{-1} \), switch the elements in each pair of R:**
- From \( R = \{(1, 5), (2, 7), (4, 11)\} \):
- The pairs become: \( (5, 1), (7, 2), (11, 4) \)

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

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

1. **Draw two circles to represent sets A and B:**
- Circle A contains elements \( \{1, 2, 3, 4\} \)
- Circle B contains elements \( \{5, 7, 8, 11, 15\} \)

2. **Draw arrows from elements in A to their corresponding elements in B based on the relation R:**
- \( 1 \rightarrow 5 \)
- \( 2 \rightarrow 7 \)
- \( 4 \rightarrow 11 \)
- Note: There is no arrow from \( 3 \) since \( 9 \) is not in B.

### Summary of the Answers:
- (i) \( R = \{(1, 5), (2, 7), (4, 11)\} \)
- (ii) Domain: \( \{1, 2, 4\} \), Range: \( \{5, 7, 11\} \)
- (iii) \( R^{-1} = \{(5, 1), (7, 2), (11, 4)\} \)
- (iv) Arrow diagram with arrows from \( 1 \) to \( 5 \), \( 2 \) to \( 7 \), and \( 4 \) to \( 11 \).