Q 29. - 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 xy-plane. 

Step-by-Step Solution

Answer

Part (a): The set is closed.

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

Part (c): The boundary of the set is the xy-plane.

Part (a): Step 1. Given information

We have given all points on the xy-plane.

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

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. It is said to be closed if its complement is open.

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

Thus, they must be an open set.

The complement of the set is all those points that do not lie in xy axis. Which is also an open set.

Thus, the set is closed because complement, $S^c$ , is open. 

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 xy plane. Because the given set is the set of all points on the xy-plane.

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

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 xy-plane.