Q. 26 - 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.

$R^2$

Step-by-Step Solution

Answer

Part (a): Both Open and Closed set 

Part (b):$\varphi$

Part (c):$R^2$

Part (a): Step 1. Given Information

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

Part (a): Step 2. Determine if 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. The set S is an all-purpose set. It involves all the points in the space of $R^2$. It does not have any boundary to identify. So, it will be an open set. The complement of  $R^2$is an empty set, which makes it  both open and closed set.

Hence, a $R^2$ is Both Open and Closed Set.

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

The goal is to find the complement of the set S. The set S refers to the all the points in the $R^2$. A set's complement is the collection of all points that aren't part of the set. The empty set is the complement of an empty set.

The compliment of S is $S^c = \varphi$

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

The goal is to find the boundary of the set S. The boundary of $R^2$ set will be $R^2$set itself. The boundary of the set S is $R^2$