Question

To describe: The representation of equation \(0 \le z \le 6\) in \({\mathbb{R}^3}\).

Short answer

The equation \(0 \le z \le 6\) in \({\mathbb{R}^3}\) represents a space that consists of all points which lie on or between the horizontal planes \(z = 0\) and \(z = 6\).

Step-by-step solution

Concept of \({\mathbb{R}^3}\)

\({\mathbb{R}^3}\)is the three dimensional coordinate system which contains\(x\),\(y\), and\(z\)-coordinates.

Represent the given equation in \({\mathbb{R}^3}\)

The equation \(0 \le z \le 6\) in \({\mathbb{R}^3}\) represents the set \(\{ (x,y,z)\mid 0 \le z \le 6\} \), which is the set of all points in \({\mathbb{R}^3}\) whose \(z\)-coordinate lies on or between 0 and 6 , and \(x\), \(y\)-coordinates are any values.

The equation \(z = 0\) is sketched as shown in Figure 1.

The equation \(z = 6\) is sketched as shown in Figure 2.

Thus, the equation \(0 \le z \le 6\) in \({\mathbb{R}^3}\) represents a space that consists of all points which lie on or between the horizontal planes \(z = 0\) and \(z = 6\).