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 \([x]\), returns the largest integer less than or equal to \(x\). For example, \([2.5] = 2\) and \([3] = 3\).

### Step 2: Set Up the Equation
From the relation, we have:
\[ [x] + [y] = 2 \]
Since both \(x\) and \(y\) are greater than 0, the possible pairs of \([x]\) and \([y]\) that satisfy this equation are:
- \([x] = 0\) and \([y] = 2\)
- \([x] = 1\) and \([y] = 1\)
- \([x] = 2\) and \([y] = 0\)

### Step 3: Determine Valid Values for \(x\) and \(y\)
1. **Case 1:** \([x] = 0\) and \([y] = 2\)
- This means \(0 \leq x < 1\) and \(2 \leq y < 3\).

2. **Case 2:** \([x] = 1\) and \([y] = 1\)
- This means \(1 \leq x < 2\) and \(1 \leq y < 2\).

3. **Case 3:** \([x] = 2\) and \([y] = 0\)
- This means \(2 \leq x < 3\) and \(0 \leq y < 1\). However, since \(y\) must be greater than 0, this case is not valid.

### Step 4: Combine the Valid Cases
From the valid cases, we have:
- From Case 1: \(0 \leq x < 1\) and \(2 \leq y < 3\)
- From Case 2: \(1 \leq x < 2\) and \(1 \leq y < 2\)

### Step 5: Determine the Domain
The domain of \(x\) is the union of the intervals from the valid cases:
- From Case 1: \(0 \leq x < 1\)
- From Case 2: \(1 \leq x < 2\)

Thus, the domain is:
\[ \text{Domain} = [0, 2) \]

### Step 6: Determine the Range
The range of \(y\) is also the union of the intervals from the valid cases:
- From Case 1: \(2 \leq y < 3\)
- From Case 2: \(1 \leq y < 2\)

Thus, the range is:
\[ \text{Range} = [1, 2) \cup [2, 3) = [1, 3) \]

### Final Answer
- **Domain:** \([0, 2)\)
- **Range:** \([1, 3)\)