Let `A={1,2,3,4,6}`. Let R be the relation on A defined by `{(a,b)=ainA,binA," a divides b"}`. Find
(i) R (ii) domain of R (iii) range of R

Answer

To solve the problem, we will follow these steps:

### Step 1: Define the relation R
Given the set \( A = \{1, 2, 3, 4, 6\} \), we define the relation \( R \) as the set of ordered pairs \( (a, b) \) such that \( a \) divides \( b \) for \( a, b \in A \).

### Step 2: Find the ordered pairs in R
We will check each element of \( A \) to see which elements it divides:

- For \( a = 1 \):
- \( 1 \div 1 = 1 \) (True)
- \( 1 \div 2 = 0.5 \) (True)
- \( 1 \div 3 = 0.33 \) (True)
- \( 1 \div 4 = 0.25 \) (True)
- \( 1 \div 6 = 0.1667 \) (True)

Thus, the pairs are: \( (1, 1), (1, 2), (1, 3), (1, 4), (1, 6) \).

- For \( a = 2 \):
- \( 2 \div 1 = 2 \) (False)
- \( 2 \div 2 = 1 \) (True)
- \( 2 \div 3 = 0.67 \) (False)
- \( 2 \div 4 = 0.5 \) (True)
- \( 2 \div 6 = 0.3333 \) (True)

Thus, the pairs are: \( (2, 2), (2, 4), (2, 6) \).

- For \( a = 3 \):
- \( 3 \div 1 = 3 \) (False)
- \( 3 \div 2 = 1.5 \) (False)
- \( 3 \div 3 = 1 \) (True)
- \( 3 \div 4 = 0.75 \) (False)
- \( 3 \div 6 = 0.5 \) (True)

Thus, the pairs are: \( (3, 3), (3, 6) \).

- For \( a = 4 \):
- \( 4 \div 1 = 4 \) (False)
- \( 4 \div 2 = 2 \) (False)
- \( 4 \div 3 = 1.33 \) (False)
- \( 4 \div 4 = 1 \) (True)
- \( 4 \div 6 = 0.6667 \) (False)

Thus, the pair is: \( (4, 4) \).

- For \( a = 6 \):
- \( 6 \div 1 = 6 \) (False)
- \( 6 \div 2 = 3 \) (False)
- \( 6 \div 3 = 2 \) (False)
- \( 6 \div 4 = 1.5 \) (False)
- \( 6 \div 6 = 1 \) (True)

Thus, the pair is: \( (6, 6) \).

### Step 3: Combine all pairs to form R
Now, we combine all the pairs we found:
\[ R = \{(1, 1), (1, 2), (1, 3), (1, 4), (1, 6), (2, 2), (2, 4), (2, 6), (3, 3), (3, 6), (4, 4), (6, 6)\} \]

### Step 4: Find the domain of R
The domain of a relation is the set of all first elements of the ordered pairs in \( R \):
\[ \text{Domain of } R = \{1, 2, 3, 4, 6\} \]

### Step 5: Find the range of R
The range of a relation is the set of all second elements of the ordered pairs in \( R \):
\[ \text{Range of } R = \{1, 2, 3, 4, 6\} \]

### Final Results
- \( R = \{(1, 1), (1, 2), (1, 3), (1, 4), (1, 6), (2, 2), (2, 4), (2, 6), (3, 3), (3, 6), (4, 4), (6, 6)\} \)
- Domain of \( R = \{1, 2, 3, 4, 6\} \)
- Range of \( R = \{1, 2, 3, 4, 6\} \)