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

The equation \(\displaystyle{z}={8}\in\mathbb{R}^{3}\) represents the set \(\displaystyle{\left\lbrace{\left({x},{y},{z}\right)}{|}{z}={8}\right\rbrace}\), which is the set of all points in \(\displaystyle\mathbb{R}^{3}\) whose z - coordinate is 8 and x, y - coordinates are any values.

The inequality to describe the region in which all points lie on or below the plane \(\displaystyle{z}={8}\ {i}{s}\ {0}\le{z}\le{8}\).

The equation to describe the region of cylinder with center at the origin and radius of 2 units on the xy-plane is \(\displaystyle{x}^{2}+{y}^{2}={4}\). But, it is required to describe the region in which all points lie on or above the disk in the xy - plane.

The inequality to describe the region of solid cylinder that lies on or above the disk in the xy - plane wiwth center at origin and radius of 2 units \(\displaystyle{x}^{2}+{y}^{2}\le{4}.\)

Thus, the inequality to describe the region of solid cylinder that lies on or below the plane \(\displaystyle{z}={8}\) and on or above the disk in the xy - plane with center at the origin and radius \(\displaystyle{2}\ {i}{s}\ {x}^{2}+{y}^{2}\le{4},{0}\le{z}\le{8}.\)

The equation \(\displaystyle{z}={8}\in\mathbb{R}^{3}\) represents the set \(\displaystyle{\left\lbrace{\left({x},{y},{z}\right)}{|}{z}={8}\right\rbrace}\), which is the set of all points in \(\displaystyle\mathbb{R}^{3}\) whose z - coordinate is 8 and x, y - coordinates are any values.

The inequality to describe the region in which all points lie on or below the plane \(\displaystyle{z}={8}\ {i}{s}\ {0}\le{z}\le{8}\).

The equation to describe the region of cylinder with center at the origin and radius of 2 units on the xy-plane is \(\displaystyle{x}^{2}+{y}^{2}={4}\). But, it is required to describe the region in which all points lie on or above the disk in the xy - plane.

The inequality to describe the region of solid cylinder that lies on or above the disk in the xy - plane wiwth center at origin and radius of 2 units \(\displaystyle{x}^{2}+{y}^{2}\le{4}.\)

Thus, the inequality to describe the region of solid cylinder that lies on or below the plane \(\displaystyle{z}={8}\) and on or above the disk in the xy - plane with center at the origin and radius \(\displaystyle{2}\ {i}{s}\ {x}^{2}+{y}^{2}\le{4},{0}\le{z}\le{8}.\)