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 (i).  .

Solution 9 (i).

See text and/or instructor's solution manual.

    First, observe that     iff   

The set    is the right half plane  .

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

                It is not closed and not bounded.

        The boundary line  x=1  is not included in the set  .

We are done.   

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.

.  This establishes that S is an unbounded 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