The area defined by [x] + [y] = 1, -1 `le` x , y `le` 3 in the x - y coordinate plane is

Options

A. 8
B. 6
C. 4
D. 2

Answer

To find the area defined by the equation \([x] + [y] = 1\) within the bounds \(-1 \leq x, y \leq 3\), we can follow these steps:

### Step 1: Understand the Greatest Integer Function
The greatest integer function \([x]\) gives the largest integer less than or equal to \(x\). For example:
- If \(-1 \leq x < 0\), then \([x] = -1\).
- If \(0 \leq x < 1\), then \([x] = 0\).
- If \(1 \leq x < 2\), then \([x] = 1\).
- If \(2 \leq x < 3\), then \([x] = 2\).
- If \(x = 3\), then \([x] = 3\).

### Step 2: Analyze the Equation \([x] + [y] = 1\)
We can break down the equation into different cases based on the values of \([x]\):

1. **Case 1**: \([x] = -1\)
- Then \([y] = 2\), which means \(2 \leq y < 3\).
- The range for \(x\) is \(-1 \leq x < 0\).
- This gives us a rectangular area defined by \((-1, 2)\) to \((0, 3)\).

2. **Case 2**: \([x] = 0\)
- Then \([y] = 1\), which means \(1 \leq y < 2\).
- The range for \(x\) is \(0 \leq x < 1\).
- This gives us another rectangular area defined by \((0, 1)\) to \((1, 2)\).

3. **Case 3**: \([x] = 1\)
- Then \([y] = 0\), which means \(0 \leq y < 1\).
- The range for \(x\) is \(1 \leq x < 2\).
- This gives us another rectangular area defined by \((1, 0)\) to \((2, 1)\).

4. **Case 4**: \([x] = 2\)
- Then \([y] = -1\), which means \(-1 \leq y < 0\).
- The range for \(x\) is \(2 \leq x < 3\).
- This gives us another rectangular area defined by \((2, -1)\) to \((3, 0)\).

### Step 3: Calculate the Area of Each Rectangle
- **Area from Case 1**: Width = 1 (from -1 to 0), Height = 1 (from 2 to 3) → Area = \(1 \times 1 = 1\)
- **Area from Case 2**: Width = 1 (from 0 to 1), Height = 1 (from 1 to 2) → Area = \(1 \times 1 = 1\)
- **Area from Case 3**: Width = 1 (from 1 to 2), Height = 1 (from 0 to 1) → Area = \(1 \times 1 = 1\)
- **Area from Case 4**: Width = 1 (from 2 to 3), Height = 1 (from -1 to 0) → Area = \(1 \times 1 = 1\)

### Step 4: Total Area
Now, we can add the areas from all cases:
\[ \text{Total Area} = 1 + 1 + 1 + 1 = 4 \]

### Final Answer
The total area defined by \([x] + [y] = 1\) within the bounds \(-1 \leq x, y \leq 3\) is \(4\) square units.

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