A point 'P' moves in xy plane in such a way that `[|x|]+[|y|]=1` where [.] denotes the greatest integer function. Area of the region representing all possible positions of the point 'P' is equal to:

Answer

To solve the problem, we need to find the area of the region in the xy-plane defined by the equation \([|x|] + [|y|] = 1\), where \([.]\) denotes the greatest integer function.

### Step-by-Step Solution:

1. **Understanding the Equation**:
The equation \([|x|] + [|y|] = 1\) implies that the sum of the greatest integer values of \(|x|\) and \(|y|\) must equal 1. This means that either \([|x|] = 0\) and \([|y|] = 1\) or \([|x|] = 1\) and \([|y|] = 0\).

2. **Finding Possible Values**:
- If \([|x|] = 0\), then \(0 \leq |x| < 1\) (which implies \(-1 < x < 1\)).
- If \([|y|] = 1\), then \(1 \leq |y| < 2\) (which implies \(1 \leq y < 2\) or \(-2 < y \leq -1\)).
- If \([|x|] = 1\), then \(1 \leq |x| < 2\) (which implies \(1 \leq x < 2\) or \(-2 < x \leq -1\)).
- If \([|y|] = 0\), then \(0 \leq |y| < 1\) (which implies \(-1 < y < 1\)).

3. **Identifying Regions**:
From the above conditions, we can identify the following regions:
- Region 1: \( -1 < x < 1 \) and \( 1 \leq y < 2 \) (First Quadrant)
- Region 2: \( -1 < x < 1 \) and \( -2 < y \leq -1 \) (Fourth Quadrant)
- Region 3: \( 1 \leq x < 2 \) and \( -1 < y < 1 \) (First Quadrant)
- Region 4: \( -2 < x \leq -1 \) and \( -1 < y < 1 \) (Second Quadrant)

4. **Graphing the Regions**:
We can graph these regions on the xy-plane. The boundaries will be:
- Vertical lines at \(x = -1\), \(x = 1\), \(x = -2\), and \(x = 2\).
- Horizontal lines at \(y = -1\), \(y = 1\), \(y = 2\), and \(y = -2\).

5. **Calculating the Area**:
Each of the identified regions forms a square:
- The square formed in the first quadrant has vertices at \((0, 1)\), \((1, 1)\), \((1, 2)\), and \((0, 2)\).
- The square in the fourth quadrant has vertices at \((0, -1)\), \((1, -1)\), \((1, -2)\), and \((0, -2)\).
- The square in the second quadrant has vertices at \((-2, 0)\), \((-1, 0)\), \((-1, -1)\), and \((-2, -1)\).
- The square in the third quadrant has vertices at \((-2, 0)\), \((-1, 0)\), \((-1, 1)\), and \((-2, 1)\).

Each square has a side length of 1, so the area of each square is \(1 \times 1 = 1\).

6. **Total Area**:
Since there are four such squares, the total area is \(4 \times 1 = 4\).

### Final Answer:
The area of the region representing all possible positions of the point 'P' is \(4\).