Q. 31 - 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.

The empty set.

Step-by-Step Solution

Answer

Part (a): The set is both open and closed.

Part (b): The compliment of the set is ${\varphi }^c=R^3$

Part (c):  The boundary of the set is $\varphi$

Part (a): Step 1: Given Information

Consider $R^3$ as a subset of the empty set.

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

Determining whether the empty set is open, closed, both open and closed, or neither open nor closed is the goal. Consider the following assertion (1): Take a look at the subset A of $R^3$ .

If there is an open disc D such that $(x,y,z)\in D\subseteq A$, then the subset A of $R^3$ is open for all $(x,y,z)\in R^3$.

Because the empty set does not include any elements, the empty set as a subset of is open, according to assertion (1). Take a look at the second statement: Consider the subset A of $R^3$.

Then the subset A of $R^3$ is closed if the compliment $A^c$ is open. The compliment set of an empty set $\varphi$ , as a subset of $R^3$ , is defined as ${\varphi }^c=\left\{\right(x,y,z)\in R^3|(x,y,z)\notin \varphi \}$ 

 As a result of assertion (1), the empty set's subset, $\varphi$, is open. As a result of assertion (2), the empty set as a subset of $R^3$ is closed. As a result, the empty set as a subset of $R^3$ is both open and closed.

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

The goal is to figure out what the set's complement is.

The compliment of the empty set $\varphi$ as a subset of $R^3$ is ${\varphi }^c=R^3$

So, the compliment of the empty set $\varphi$ as a subset of $R^3$ is ${\varphi }^c=R^3$.

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

The goal is to determine the empty set's border. Consider the following: (3) Consider $R^3$'s subset A. The point $(x,y,z)$ is thus said to be a boundary point A, if the open ball in $R^3$ contacting $(x,y,z)$ intersects both $A and A^c$.

As a result of assertion , the border of the empty set $\varphi$ is $\varphi$.