If set A=`{1,2,3,4}` and a relation R is defined from A to A as follows:
`R={(x,y): x gt 1 , y=3}`
Find the domain and range of R.

Answer

To solve the problem, we need to analyze the given relation \( R \) defined from set \( A \) to set \( A \).

### Step-by-Step Solution:

1. **Identify the Set A**:
The set \( A \) is given as \( A = \{1, 2, 3, 4\} \).

2. **Understand the Relation R**:
The relation \( R \) is defined as:
\[ R = \{(x, y) : x > 1, y = 3\} \]
This means that for every \( x \) that is greater than 1, \( y \) must be equal to 3.

3. **Determine Possible Values for x**:
Since \( x \) must be greater than 1, we look at the elements of set \( A \):
- The elements of \( A \) are \( 1, 2, 3, 4 \).
- The values of \( x \) that satisfy \( x > 1 \) are \( 2, 3, 4 \).

4. **Determine the Fixed Value of y**:
According to the relation, \( y \) is fixed at 3. Therefore, for each valid \( x \), \( y \) will always be 3.

5. **Construct the Relation R**:
Based on the values of \( x \) and the fixed value of \( y \), we can construct the relation \( R \):
\[ R = \{(2, 3), (3, 3), (4, 3)\} \]

6. **Find the Domain of R**:
The domain of a relation is the set of all first elements (x-values) in the ordered pairs. From our relation \( R \):
- The x-values are \( 2, 3, 4 \).
- Therefore, the domain is:
\[ \text{Domain} = \{2, 3, 4\} \]

7. **Find the Range of R**:
The range of a relation is the set of all second elements (y-values) in the ordered pairs. From our relation \( R \):
- The y-value is always 3 for all pairs.
- Therefore, the range is:
\[ \text{Range} = \{3\} \]

### Final Answer:
- **Domain of R**: \( \{2, 3, 4\} \)
- **Range of R**: \( \{3\} \)