If `A={2,4,6,9},B={4,6,18,27,54}` and a relation `R` from `A` to `B` is defined by `R=(a,b):a in A ,b in B ,`a is a factor of b` and a lt b}`,then find in roaster form .Also find its domain and range.

Answer

To solve the problem, we need to find the relation \( R \) from set \( A \) to set \( B \) defined by the conditions that \( a \) is a factor of \( b \) and \( a < b \).

### Step-by-Step Solution:

1. **Identify the Sets**:
- Let \( A = \{2, 4, 6, 9\} \)
- Let \( B = \{4, 6, 18, 27, 54\} \)

2. **Define the Relation**:
- The relation \( R \) consists of ordered pairs \( (a, b) \) such that:
- \( a \in A \)
- \( b \in B \)
- \( a \) is a factor of \( b \)
- \( a < b \)

3. **Check Each Element of \( A \) Against \( B \)**:
- For \( a = 2 \):
- \( b = 4 \): \( 2 \) is a factor of \( 4 \) and \( 2 < 4 \) → Include \( (2, 4) \)
- \( b = 6 \): \( 2 \) is a factor of \( 6 \) and \( 2 < 6 \) → Include \( (2, 6) \)
- \( b = 18 \): \( 2 \) is a factor of \( 18 \) and \( 2 < 18 \) → Include \( (2, 18) \)
- \( b = 27 \): \( 2 \) is a factor of \( 27 \) but \( 2 < 27 \) is not applicable → Exclude
- \( b = 54 \): \( 2 \) is a factor of \( 54 \) and \( 2 < 54 \) → Include \( (2, 54) \)

- For \( a = 4 \):
- \( b = 4 \): \( 4 \) is not less than \( 4 \) → Exclude
- \( b = 6 \): \( 4 \) is not a factor of \( 6 \) → Exclude
- \( b = 18 \): \( 4 \) is not a factor of \( 18 \) → Exclude
- \( b = 27 \): \( 4 \) is not a factor of \( 27 \) → Exclude
- \( b = 54 \): \( 4 \) is not a factor of \( 54 \) → Exclude

- For \( a = 6 \):
- \( b = 4 \): \( 6 \) is not a factor of \( 4 \) → Exclude
- \( b = 6 \): \( 6 \) is not less than \( 6 \) → Exclude
- \( b = 18 \): \( 6 \) is a factor of \( 18 \) and \( 6 < 18 \) → Include \( (6, 18) \)
- \( b = 27 \): \( 6 \) is not a factor of \( 27 \) → Exclude
- \( b = 54 \): \( 6 \) is a factor of \( 54 \) and \( 6 < 54 \) → Include \( (6, 54) \)

- For \( a = 9 \):
- \( b = 4 \): \( 9 \) is not a factor of \( 4 \) → Exclude
- \( b = 6 \): \( 9 \) is not a factor of \( 6 \) → Exclude
- \( b = 18 \): \( 9 \) is a factor of \( 18 \) and \( 9 < 18 \) → Include \( (9, 18) \)
- \( b = 27 \): \( 9 \) is a factor of \( 27 \) and \( 9 < 27 \) → Include \( (9, 27) \)
- \( b = 54 \): \( 9 \) is a factor of \( 54 \) and \( 9 < 54 \) → Include \( (9, 54) \)

4. **Compile the Relation**:
- The relation \( R \) in roaster form is:
\[ R = \{(2, 4), (2, 6), (2, 18), (2, 54), (6, 18), (6, 54), (9, 18), (9, 27), (9, 54)\} \]

5. **Determine the Domain and Range**:
- **Domain**: The set of all first elements (from set \( A \)) in the ordered pairs of \( R \):
\[ \text{Domain} = \{2, 6, 9\} \]
- **Range**: The set of all second elements (from set \( B \)) in the ordered pairs of \( R \):
\[ \text{Range} = \{4, 6, 18, 27, 54\} \]

### Final Answer:
- Relation \( R \) in roaster form:
\[ R = \{(2, 4), (2, 6), (2, 18), (2, 54), (6, 18), (6, 54), (9, 18), (9, 27), (9, 54)\} \]
- Domain:
\[ \{2, 6, 9\} \]
- Range:
\[ \{4, 6, 18, 27, 54\} \]