Exercise 9.  Consider the following sets.   Sketch each set.   State, with reasons, which of the following terms apply to the above sets: open; connected; domain; region; closed region; bounded.  

9 (v).  .

Solution 9 (v).

See text and/or instructor's solution manual.

    The set    is the portion of the unit disk    that lies in the right half plane  .

                It is an open connected domain.  It is also a region.  It is bounded.  

                It is not closed.

        The boundary semi-circle    and the segment    are not included
        in the set  .

We are done.   

The boundary circle    and the imaginary axis are not included in the set  .

For each    there exists an    so that     (choose  ).  This argument establishes that S is an open set.

Every pair of points in S can be connected with a straight line.  This argument establishes that S is a connected set.

Aside.  We can let Mathematica double check our work.

This solution is complements of the authors.

(c) 2008 John H. Mathews, Russell W. Howell