Q 28. - Calculus

Question

In Exercises 27–32, (a) determine whether the given subset of $R^3$ 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) Set S is both open and closed.

Part (b) The complement of the set is $S^c=\left\{\right(x,y,z\left)\right|x,y,z\ne 0\}$

Part (c) The boundary of the set is $\left\{\right(x,y,0\left)\right|x,y\in R\}\cup \{(x,0,z)|x,z\in R\}\cup \left\{\right(0,y,z\left)\right|y,z\in R\}$

Part (a): Step 1. Given information

We have given all points on the coordinate axes. 

The set can be written as: $\left\{\right(x,y,0\left)\right|x,y\in R\}\cup \{(x,0,z)|x,z\in R\}\cup \left\{\right(0,y,z\left)\right|y,z\in R\}$

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

We know that subset is open if it does not have a boundary.

In the given set, all the points lie on the coordinate axes. It means there is no limit on them.

Thus, they do not provide any boundary condition so they must be an open set.

The complement of the set is also an open set, hence it follows the definition of the closed set.

Thus, the set is both open and closed.

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

The complement of the set is the collection of all the points which do not lie on the coordinate axes because the given set is the set of all the points on the coordinate axes.

So, the complement is $S^c=\left\{\right(x,y,z\left)\right|x,y,z\ne 0\}$

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

The boundary of the set is the extreme values of the variables involved.

Thus, the boundary of the set S is the positive axes that are, $\left\{\right(x,y,0\left)\right|x,y\in R\}\cup \{(x,0,z)|x,z\in R\}\cup \left\{\right(0,y,z\left)\right|y,z\in R\}$