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 will follow these steps:

### Step 1: Understand the Greatest Integer Function
The greatest integer function \([x]\) gives the largest integer less than or equal to \(x\). We need to analyze the values of \([x]\) and \([y]\) within the given range.

### Step 2: Determine the Possible Values of \([x]\) and \([y]\)
Given the range \(-1 \leq x, y \leq 3\):
- The possible values for \([x]\) are \(-1, 0, 1, 2, 3\).
- The possible values for \([y]\) are also \(-1, 0, 1, 2, 3\).

### Step 3: Solve the Equation \([x] + [y] = 1\)
Now we will find the pairs \(([x], [y])\) that satisfy the equation \([x] + [y] = 1\):
1. If \([x] = -1\), then \([y] = 2\).
2. If \([x] = 0\), then \([y] = 1\).
3. If \([x] = 1\), then \([y] = 0\).
4. If \([x] = 2\), then \([y] = -1\).

### Step 4: Determine the Ranges for \(x\) and \(y\)
From the pairs found:
1. For \([x] = -1\) and \([y] = 2\):
- \(x \in [-1, 0)\) and \(y \in [2, 3)\) → Area is a rectangle with width 1 and height 1.
2. For \([x] = 0\) and \([y] = 1\):
- \(x \in [0, 1)\) and \(y \in [1, 2)\) → Area is a rectangle with width 1 and height 1.
3. For \([x] = 1\) and \([y] = 0\):
- \(x \in [1, 2)\) and \(y \in [0, 1)\) → Area is a rectangle with width 1 and height 1.
4. For \([x] = 2\) and \([y] = -1\):
- \(x \in [2, 3)\) and \(y \in [-1, 0)\) → Area is a rectangle with width 1 and height 1.

### Step 5: Calculate the Total Area
Each of the above pairs gives an area of \(1 \times 1 = 1\) square unit. Since there are 4 such rectangles, the total area is:
\[ \text{Total Area} = 4 \times 1 = 4 \text{ square units} \]

### Final Answer
The area defined by \([x] + [y] = 1\) in the given bounds is \(4\) square units.

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