Find area of the region in the xy plane where

[x][y]=[x] + [y] + 1 where [.] denotes greatest integer function

To find the area of the region in the xy-plane defined by the equation \([x][y] = [x] + [y] + 1\), where \([.]\) denotes the greatest integer function (also known as the floor function), we need to analyze the implications of this equation step by step.

Understanding the Greatest Integer Function

The greatest integer function, \([x]\), gives the largest integer less than or equal to \(x\). For example, \([2.7] = 2\) and \([-1.3] = -2\). This function creates a piecewise constant behavior, which is crucial for our analysis.

Breaking Down the Equation

The equation can be rewritten as:

\([x][y] - [x] - [y] = 1\

Let’s denote \([x] = m\) and \([y] = n\), where \(m\) and \(n\) are integers. The equation then transforms into:

\(mn - m - n = 1\)

Rearranging the Equation

We can rearrange this equation to:

\(mn - m - n + 1 = 2\)

This can be factored as:

\((m-1)(n-1) = 2\)

Finding Integer Solutions

Next, we need to find pairs of integers \((m-1)\) and \((n-1)\) that multiply to 2. The integer factor pairs of 2 are:

• (1, 2)
• (2, 1)
• (-1, -2)
• (-2, -1)

Calculating Values of m and n

From these pairs, we can derive the following values for \(m\) and \(n\):

• From (1, 2): \(m = 2\), \(n = 3\)
• From (2, 1): \(m = 3\), \(n = 2\)
• From (-1, -2): \(m = 0\), \(n = -1\)
• From (-2, -1): \(m = -1\), \(n = 0\)

Identifying the Regions

Now, we can summarize the integer pairs \((m, n)\) that satisfy the equation:

• (2, 3)
• (3, 2)
• (0, -1)
• (-1, 0)

Mapping to the xy-plane

Each pair \((m, n)\) corresponds to a rectangular region in the xy-plane:

• For \(m = 2\) and \(n = 3\): The region is \(2 \leq x < 3\) and \(3 \leq y < 4\)
• For \(m = 3\) and \(n = 2\): The region is \(3 \leq x < 4\) and \(2 \leq y < 3\)
• For \(m = 0\) and \(n = -1\): The region is \(0 \leq x < 1\) and \(-1 \leq y < 0\)
• For \(m = -1\) and \(n = 0\): The region is \(-1 \leq x < 0\) and \(0 \leq y < 1\)

Calculating the Area

Now, let’s calculate the area of each rectangular region:

• Area of region for (2, 3): Width = 1, Height = 1 → Area = 1
• Area of region for (3, 2): Width = 1, Height = 1 → Area = 1
• Area of region for (0, -1): Width = 1, Height = 1 → Area = 1
• Area of region for (-1, 0): Width = 1, Height = 1 → Area = 1

Summing Up the Areas

The total area of the regions is:

Area = 1 + 1 + 1 + 1 = 4

Thus, the area of the region in the xy-plane defined by the equation \([x][y] = [x] + [y] + 1\) is 4 square units.