Q. 27 - 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 $x>0$,$y<0$ and $z<0$

Step-by-Step Solution

Answer

Part (a): Open Set

Part (b): $\left\{\right(x,y,z) | x\le 0 OR y\ge 0 OR z \ge 0\}$

Part (c): $\left\{\right(x,y,0) | x\ge 0,y\le 0\}\cup \{(x,0,z) | x\ge 0,z\le 0\}\cup \left\{\right(0,y,z) | y\le 0,z\le 0\}$

Part (a): Step 1. Given Information

Consider S is a subset of $R^3$is defined as follows :

$S =\left\{\right(x,y,z) | x>0,y<0,z<0\}$

Part (a): Step 2. Determine if the the set is open, closed, both open and closed, or neither open nor closed.

The goal is to figure out if set S is open, closed, open and closed, or neither open nor closed. If there is no boundary to identify, a subset is said to be open. All of the inequalities in set S are of the type "greater than" and "less than." As a result, the set has no boundaries. 

Hence, this set is an Open Set

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

The goal is to discover the complement of the set S. All of the points in the first quadrant are included in the set S in $xy$-plane. A set's complement is the collection of all points that aren't part of the set. As a result, the points in the other three quadrants are the complement of set S in $xy$-plane.

The compliment of the set S is $\left\{\right(x,y,z) | x\le 0 OR y\ge 0 OR z \ge 0\}$

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

The goal is to determine the set S's border. The extreme values of the variables involved form the set's border.

The boundary of the set s is 

 $\left\{\right(x,y,0) | x\ge 0,y\le 0\}\cup \{(x,0,z) | x\ge 0,z\le 0\}\cup \left\{\right(0,y,z) | y\le 0,z\le 0\}$