Q. 23 - 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 the points satisfying the inequality $\left|x\right| + \left|y\right| \le 1$

Step-by-Step Solution

Answer

Part (a): Closed Set

Part (b): $S^c = \left\{\right(x,y) | \left|x\right|+\left|y\right| = 1\}$

Part (c): $\left\{\right(x,y) | \left|x\right|+\left|y\right| = 1\}$

Part (a): Step 1. Given Information

Consider S is a subject of $R^2$ and is define as follows $\left\{\right(x,y) | \left|x\right|+\left|y\right| = 1\}$

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, both open and closed, or neither open nor closed. If there is no boundary to identify, a subset is said to be open. The inequality in the set S is of the kind "less than or equal to." As a result, the border is firmly defined.

Hence, the set S is 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. As a result, the points that do not meet this inequality form the complement of set S.

The compliment of the set S is $S^c = \left\{\right(x,y) | \left|x\right|+\left|y\right| = 1\}$

Part (c): Finding the boundary of the given 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,

$\left\{\right(x,y) | \left|x\right|+\left|y\right| = 1\}$