Q. 30 - Calculus

Question

In Exercises 27–32, (a) determine whether the given subset of $R^3$ is open, closed, both open and closed, or neither open nor closed, (b) find the complement of the set, and (c) find the boundary of the given set.

All points $(x,y,z) such that \left|x\right|<1,\left|y\right|<1, and \left|z\right|>3$.

Step-by-Step Solution

Answer

Part (a): The set is an open set

Part (b): The compliment of the set is $S^c=\left\{\right(x,y,z\left)\right|\left|x\right|\ge 1,\left|y\right|\ge 2,\left|z\right|\le 3\}$

Part (c): The boundary of the set is $\left\{\right(x,y,z\left)\right|\left|x\right|=1,\left|y\right|=2,\left|z\right|=3\}$

Part (a): Step 1: Given Information.

Consider the following definition of S as a subset of $R^3$ is defined as:

 $S=\left\{\right(x,y,z\left)\right|\left|x\right|>1,\left|y\right|>2,\left|z\right|<3\}$ 

Part (a): Step 2: Identify whether the set is open, closed both, or neither.

The goal is to figure out if the set S is open, closed, both open and closed, or not open at all. If a subset does not have a defined border, it is said to be open. All inequalities in the set S are open. As a result, the set S stands for Open Set.

Part (b): Step 1: Determine the complement of the set

The goal is to discover the set S's complement. All points on the coordinate axes are referred to as the set S. A set's complement is the collection of all points that do not belong to it. The points that do not meet this inequality make up the complement of set S.

The complement of set S is $S^c=\left\{\right(x,y,z\left)\right|\left|x\right|\ge 1,\left|y\right|\ge 2,\left|z\right|\le 3\}$

Part (c): Step 1: Determine the boundary of the set.

The goal is to determine the set S's border.

The extreme values of the variables involved form the set's border. As a result, the set S's boundary is $\left\{\right(x,y,z\left)\right|\left|x\right|=1,\left|y\right|=2,\left|z\right|=3\}$