Q. 21 - Calculus

Question

In Exercises 21–26, (a) determine whether the given subset of $R^2$ 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 $(x,y)$ such that $x >0$ and $y >0$

Step-by-Step Solution

Answer

Part (a): Open Set

Part (b):$S^c= \left\{\right(x,y) | x \le 0 or y \le 0\}$

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

Part (a): Step 1. Given Information

Consider S is a subset of $R^2$ and is defined as follows:

$S^c= \left\{\right(x,y) | x \le 0 or y \le 0\}$

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

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. Both inequalities in the set S are of the "greater than" kind. As a result, they supply no boundary conditions.

As a result, this is an Open set.

Part (b): Step 1 Given Information

Consider S is a subset of $R^2$ and is defined as follows:$S^c= \left\{\right(x,y) | x \le 0 or y \le 0\}$

The objective is to find the complement of the set S. 

Part (b): Step 2. Finding compliment of the set

All of the points in the first quarter of a $xy$-plane are included in the set S. A set's complement is the collection of all points that aren't part of the set. The points in the other three quadrants of the $xy$-plane are thus the complement of the set S.

The complement of given set S is 

$S^c= \left\{\right(x,y) | x \le 0 or y \le 0\}$

Part (c): Step 1 Given Information

Consider S is a subset of $R^2$and is defined as follows:

$S^c= \left\{\right(x,y) | x \le 0 or y \le 0\}$

The objective is to find the boundary of the set S. 

Part (c): Step 2. Finding the boundary of the set

The extreme values of the variables involved form the set's border. The positive axis are therefore the set S's border. These boundary conditions can be expressed as sets, such as,

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