Q. 32 - 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.

$R^3$

Step-by-Step Solution

Answer

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

Part (b): The compliment of the set is $\varphi$

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

Part (a): Step 1. Given Information

The goal is to figure out if the set $R^3$ is open, closed, open and closed, or neither open nor closed.

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

Take a look at the following statement (1): Consider the A subset of $R^3$ 

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

Since there exists an open disc $D_1$ for each element $(a,b,c)\in R^3$ such that $(x,y,z)\in D_1\subseteq R^3$ . As a result of assertion (1), the set $R^3$ as a subset of $R^3$ is open

Take a look at the second statement: Consider the subset A of $R^3$.

 If its complement, $A_c$, is open, then the subset A of $R^3$ is closed.

The compliment of the set $R^3$ as subset of $R^3$ is defined as

${\left\{R^3\right\}}^C= \left\{\right(x,y,z)\in R^3|(x,y,z)\in R^3\}$

As a result, the complement of the set $R^3$, as a subset of $R^{3 }is {\left(R^3\right)}^c=\varphi$

As a result of assertion (1), the empty set $\varphi$ is open.

 As a result of assertion (2), the set $R^3$ as a subset of  $R^3$ is closed.

As a result, the set $R^3$ 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 find the complement of the set $R^3$ as a subset of $R^3$. The complement of the set $R^3$ as a subset of $R^3$ is defined as ${\left(R^3\right)}^C=\left\{\right(x,y,z)\in R^3|(x,y,z)\in R^3\}$ 

As a result, the complement of the set $R^3$ as a subset of $R^3$ is ${\left(R^3\right)}^C=\varphi$.

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

The goal is to discover the border of set $R^3$ as a subset of $R^3$

Consider the following statement (3):

Consider the subset A of $R^3$.

The point $(x,y,z)$ is said to be a boundary point of A if the open ball in the $R^3$ containing $(x,y,z)$ crosses both $A and A^c$.

Consider the point $(a,b,c)\in R^3$

Consider an open ball $S_1$ containing $(a,b,c)$.

As a result, the open ball $S_1$ intersects $R^3$ but not ${\left(R^3\right)}^C=\varphi$.

Thus, $(a,b,c)\in R^3$ is not the border point of set $R^3$.

Because $(a,b,c)\in R^3$ is arbitrary, the boundary of the set $R^3$ as a subset of $R^3 is \varphi$.