Domain and range of the realtion `R={(x,y) : [x]+[y]=2, x gt 0, y gt 0}` are

Options

A. `[0,3],[0,3]`
B. `[0,3),[0,3)`
C. `(0,3),(0,3)`
D. `[0,3],[0,3)`

Answer

To find the domain and range of the relation \( R = \{(x, y) : [x] + [y] = 2, x > 0, y > 0\} \), we will follow these steps:

### Step 1: Understand the Greatest Integer Function
The greatest integer function, denoted as \([x]\), gives the largest integer less than or equal to \(x\). For example, \([2.5] = 2\) and \([3] = 3\).

### Step 2: Set Up the Equation
Given the equation \([x] + [y] = 2\), we need to find pairs of integers \([x]\) and \([y]\) that satisfy this equation. The possible pairs of integers that add up to 2 are:
- \([x] = 0\) and \([y] = 2\)
- \([x] = 1\) and \([y] = 1\)
- \([x] = 2\) and \([y] = 0\)

### Step 3: Analyze Each Case
1. **Case 1:** \([x] = 0\) and \([y] = 2\)
- This implies \(0 \leq x < 1\) and \(2 \leq y < 3\).
- Since \(x > 0\), we have \(0 < x < 1\) and \(2 \leq y < 3\).

2. **Case 2:** \([x] = 1\) and \([y] = 1\)
- This implies \(1 \leq x < 2\) and \(1 \leq y < 2\).
- Both \(x\) and \(y\) are in the range \(1 \leq x < 2\) and \(1 \leq y < 2\).

3. **Case 3:** \([x] = 2\) and \([y] = 0\)
- This implies \(2 \leq x < 3\) and \(0 \leq y < 1\).
- Since \(y > 0\), we have \(0 < y < 1\) and \(2 \leq x < 3\).

### Step 4: Combine the Results
Now, we combine the ranges from all cases:
- From Case 1: \(0 < x < 1\) and \(2 \leq y < 3\)
- From Case 2: \(1 \leq x < 2\) and \(1 \leq y < 2\)
- From Case 3: \(2 \leq x < 3\) and \(0 < y < 1\)

### Step 5: Determine the Domain
The domain consists of all possible values of \(x\):
- From Case 1: \(0 < x < 1\)
- From Case 2: \(1 \leq x < 2\)
- From Case 3: \(2 \leq x < 3\)

Thus, the domain is:
\[ (0, 3) \]

### Step 6: Determine the Range
The range consists of all possible values of \(y\):
- From Case 1: \(2 \leq y < 3\)
- From Case 2: \(1 \leq y < 2\)
- From Case 3: \(0 < y < 1\)

Thus, the range is:
\[ (0, 3) \]

### Final Answer
The domain and range of the relation \( R \) are both:
\[ (0, 3) \]