Let B = {1, 2, 3, 4, 5, .....30}, A = {2,3, 4, 5, 6, 7, 8, 9, 10}. A relation is defined from A to B defined by R ={(a, b) : b = 4a, `a in A, b in B`}, then domain and range

Answer

To solve the problem, we need to find the domain and range of the relation \( R \) defined from set \( A \) to set \( B \) by the rule \( b = 4a \).

### Step-by-Step Solution:

1. **Identify the Sets**:
- Set \( A = \{2, 3, 4, 5, 6, 7, 8, 9, 10\} \)
- Set \( B = \{1, 2, 3, \ldots, 30\} \)

2. **Define the Relation**:
- The relation \( R \) is defined as \( R = \{(a, b) : b = 4a, a \in A, b \in B\} \).

3. **Calculate the Values of \( b \)**:
- We will calculate \( b \) for each \( a \) in set \( A \):
- For \( a = 2 \): \( b = 4 \times 2 = 8 \)
- For \( a = 3 \): \( b = 4 \times 3 = 12 \)
- For \( a = 4 \): \( b = 4 \times 4 = 16 \)
- For \( a = 5 \): \( b = 4 \times 5 = 20 \)
- For \( a = 6 \): \( b = 4 \times 6 = 24 \)
- For \( a = 7 \): \( b = 4 \times 7 = 28 \)
- For \( a = 8 \): \( b = 4 \times 8 = 32 \) (not in \( B \))
- For \( a = 9 \): \( b = 4 \times 9 = 36 \) (not in \( B \))
- For \( a = 10 \): \( b = 4 \times 10 = 40 \) (not in \( B \))

4. **Determine the Domain**:
- The domain consists of all values of \( a \) for which \( b \) is in set \( B \):
- Valid pairs: \( (2, 8), (3, 12), (4, 16), (5, 20), (6, 24), (7, 28) \)
- Therefore, the domain is \( \{2, 3, 4, 5, 6, 7\} \).

5. **Determine the Range**:
- The range consists of all values of \( b \) that correspond to the valid \( a \):
- From the valid pairs, the range is \( \{8, 12, 16, 20, 24, 28\} \).

### Final Answer:
- **Domain**: \( \{2, 3, 4, 5, 6, 7\} \)
- **Range**: \( \{8, 12, 16, 20, 24, 28\} \)