Let A = {2, 3, 4}, B ={3, 4, 5} be two sets and a relation R is defined from A to B is such that R ={(x, y) : x divides y, `x in A, y in B`} then domain of R is

Options

A. {2, 3, 4}
B. {2, 3} only
C. {2, 4} only
D. {4, 3} only

Answer

To find the domain of the relation \( R \) defined from set \( A \) to set \( B \), we follow these steps:

### Step 1: Identify the sets
We have two sets:
- \( A = \{2, 3, 4\} \)
- \( B = \{3, 4, 5\} \)

### Step 2: Define the relation
The relation \( R \) is defined as:
\[ R = \{(x, y) : x \text{ divides } y, \, x \in A, \, y \in B\} \]
This means we need to find pairs \( (x, y) \) such that \( x \) divides \( y \).

### Step 3: Check each element of \( A \) against elements of \( B \)

1. **For \( x = 2 \)**:
- Check \( y = 3 \): \( 2 \) does not divide \( 3 \).
- Check \( y = 4 \): \( 2 \) divides \( 4 \) (since \( 4 \div 2 = 2 \)).
- Check \( y = 5 \): \( 2 \) does not divide \( 5 \).
- Thus, we have the pair \( (2, 4) \).

2. **For \( x = 3 \)**:
- Check \( y = 3 \): \( 3 \) divides \( 3 \) (since \( 3 \div 3 = 1 \)).
- Check \( y = 4 \): \( 3 \) does not divide \( 4 \).
- Check \( y = 5 \): \( 3 \) does not divide \( 5 \).
- Thus, we have the pair \( (3, 3) \).

3. **For \( x = 4 \)**:
- Check \( y = 3 \): \( 4 \) does not divide \( 3 \).
- Check \( y = 4 \): \( 4 \) divides \( 4 \) (since \( 4 \div 4 = 1 \)).
- Check \( y = 5 \): \( 4 \) does not divide \( 5 \).
- Thus, we have the pair \( (4, 4) \).

### Step 4: Collect all valid \( x \) values
From our checks, we found the following pairs:
- \( (2, 4) \)
- \( (3, 3) \)
- \( (4, 4) \)

The domain of the relation \( R \) consists of all \( x \) values from these pairs:
- The domain is \( \{2, 3, 4\} \).

### Final Answer
Thus, the domain of \( R \) is:
\[ \text{Domain of } R = \{2, 3, 4\} \]

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