Q 24. - 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 $\left|x\right|<1$ and $\left|y\right|\le 2$

Step-by-Step Solution

Answer

Part (a): Neither Open nor Closed Set

Part (b): $\left\{\right(x,y) | \left|x\right|>1,\left|y\right|>2 \}$

Part (c): $\left\{\right(x,y) | \left|x\right|=1,\left|y\right|=2\}$

Part (a): Step 1. Given Information

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

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 first inequality in the set S is open, whereas the second inequality is closed.

Hence, this set is neither Open nor 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 are the complement of set S.

The compliment of the set S is $\left\{\right(x,y) | \left|x\right|>1,\left|y\right|>2 \}$

Part (c): Step 1. 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,

$\left\{\right(x,y) | \left|x\right|=1,\left|y\right|=2\}$