Below is the preliminary version of Project IV ", which was posted online at about the same time as Project IV ' was posted.
Here is the link to the final version (I hope) of Project IV ".
Throughout the following, the letters TBA stand for To Be Announced, and the letters TBC stand for To Be Continued.

1. TBA

Comment:
Project IV  and Project IV '  addressed the question of whether the product of odd functions is odd. Of course, this is NOT always true, and, in the spirit of Part I of Project IV, you should prove that this is not always true by giving a counterexample:  I.e., by giving two odd functions whose product is not odd.

Since Part 2 of the Project was framed in terms of a given nonzero vector space (which was called  V),  the most complete and satisfying form of counterexample would be one which works for an arbitrary vector space. It is at least conceivable that there might be vector spaces for which the product of odd functions IS always odd.

2. Give an example of a vector space for which the product of odd functions is always odd.

Most students chose, often without pointing this out explicitly, to give a counterexample only for the case where the vector space is the set of real numbers (with the usual operations of addition and multiplication). A typical counterexample started as follows:

Choose functions  f  and  g by letting  f(x) = g(x) = TBA.

3. Explain what essential ingredient is missing from this definition of the functions. (This has nothing to do with vector spaces; it is about the polite way to define functions and introduce variables.)
4. TBA

We have now chosen functions to be used as counterexamples. ...

TBC

Last modified Apr 16, 2008 6:54 PM

Return to the Course Home Page.