Q 22 - 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 on the coordinate axes

Step-by-Step Solution

Answer

Part (a): Both Open and Closed set

Part (b): $\left\{\right(x,y) | x,y \ne 0\}$

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

Part (a): Step 1. Given Information

Consider S is a subset of $R^2$ and is define as follows $\left\{\right(x,0) | x \in R\} \cup \{(0,y) | y\in R\}$

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

The goal is to determine if set S is open. open, closed, or neither open nor closed.

If there is no boundary to identify, a subset is said to be open. Both points in the set S are on the coordinate axes. As a result, they are unrestricted. As a result, they supply no boundary conditions. As a result, the set looks to be open. The set S's counterpart is likewise an open set. Hence, it also follows the notion of closed set.

Hence, the set S  is both Open and Closed 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 on the coordinate axes are referred to as the set S. A set's complement is the collection of all points that aren't part of the set. The points that do not lie on the coordinate axes are thus the complement of set S.

The compliment of the given set S is $\left\{\right(x,y) | x,y \ne 0\}$

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 positive axis are therefore the set S's border. These boundary conditions can be expressed as sets as follows:

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