A relation `R` is defined from {2, 3, 4, 5} to {3, 6, 7, 10} by : `x\ R\ yhArrx` is relatively prime to `ydot` Then, domain of `R` is (a) {2, 3, 5} (b) {3, 5} (c) {2, 3, 4} (d) {2, 3, 4, 5}

Explanation

To determine the domain of the relation \( R \) defined from the set \( \{2, 3, 4, 5\} \) to the set \( \{3, 6, 7, 10\} \) where \( x \, R \, y \) if and only if \( x \) is relatively prime to \( y \), we will follow these steps:

### Step-by-Step Solution:

1. **Understand the Definition of Relatively Prime**:
Two numbers \( x \) and \( y \) are relatively prime if their greatest common divisor (GCD) is 1. This means they do not share any prime factors.

2. **List the Elements**:
The domain of \( R \) is the set \( \{2, 3, 4, 5\} \).

3. **Check Each Element Against the Codomain**:
We will check each element in the domain against each element in the codomain \( \{3, 6, 7, 10\} \) to see if they are relatively prime.

- For \( x = 2 \):
- \( \text{GCD}(2, 3) = 1 \) (relatively prime)
- \( \text{GCD}(2, 6) = 2 \) (not relatively prime)
- \( \text{GCD}(2, 7) = 1 \) (relatively prime)
- \( \text{GCD}(2, 10) = 2 \) (not relatively prime)

- For \( x = 3 \):
- \( \text{GCD}(3, 3) = 3 \) (not relatively prime)
- \( \text{GCD}(3, 6) = 3 \) (not relatively prime)
- \( \text{GCD}(3, 7) = 1 \) (relatively prime)
- \( \text{GCD}(3, 10) = 1 \) (relatively prime)

- For \( x = 4 \):
- \( \text{GCD}(4, 3) = 1 \) (relatively prime)
- \( \text{GCD}(4, 6) = 2 \) (not relatively prime)
- \( \text{GCD}(4, 7) = 1 \) (relatively prime)
- \( \text{GCD}(4, 10) = 2 \) (not relatively prime)

- For \( x = 5 \):
- \( \text{GCD}(5, 3) = 1 \) (relatively prime)
- \( \text{GCD}(5, 6) = 1 \) (relatively prime)
- \( \text{GCD}(5, 7) = 1 \) (relatively prime)
- \( \text{GCD}(5, 10) = 5 \) (not relatively prime)

4. **Compile the Results**:
The elements from the domain \( \{2, 3, 4, 5\} \) that are relatively prime to at least one element in the codomain \( \{3, 6, 7, 10\} \) are:
- \( 2 \) (relatively prime to \( 3 \) and \( 7 \))
- \( 3 \) (relatively prime to \( 7 \) and \( 10 \))
- \( 4 \) (relatively prime to \( 3 \) and \( 7 \))
- \( 5 \) (relatively prime to \( 3, 6, 7 \))

5. **Determine the Domain of \( R \)**:
Since all elements \( 2, 3, 4, 5 \) are related to at least one element in the codomain, the domain of the relation \( R \) is \( \{2, 3, 4, 5\} \).

### Final Answer:
The domain of \( R \) is \( \{2, 3, 4, 5\} \), which corresponds to option (d).

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