Find a counterexample.
I.e., find an element, say  s,  in  S  such that  p(s)  is not true.
To PROVE that the counterexample is really a counterexample, one should

• Exhibit an  s  which is going to be used as a counterexample.
• Prove that this  s  is in  S,  if this is not obvious.
• Prove that  p(x)  is not true,  if this is not obvious.)

Part II. (See additional Notes.) Do Problem 1 below for odd functions and cool functions (you can assume that the statement given about even functions is true).  Prove all your assertions carefully.  E.g., if you claim a function is odd, prove it. Etc.

We consider functions from a vector space  V  to the set of real numbers. (Do not get hung up on the fact that the domain is a vector space. If this causes you any trouble, you will lose only a few points if you assume that the domain is the set of real numbers.  But if you do anything under the assumption that the domain of the functions being considered is NOT a general vector space, be sure to point out explicitly what assumption(s) you are making!) The product,  fg,  of two such functions  f  and  g  is defined in the way that a product of functions is usually defined in algebra and calculus.  If  f  is such a function, we say that the function  f  is

• even  iff, for every  x  in  V,   f(-x) = f(x).
• odd  iff, for every  x  in  V,   f(-x) = -f(x).
• cool  iff it is both even and odd.

Comment:  The concepts of “even” and “odd” functions are standard and should be familiar. The concept of “cool” functions is specific for this Project, and this is not standard terminology.

1. Let  V  be a given nonzero vector space. Prove or disprove each of the following statements (give complete, careful proofs, using the standard method for proving universally quantified statements; see Part I):
   1. The product of two* even functions is even.
   2. The product of two* odd functions is odd.
   3. The product of two* cool functions is cool.
   *We use the word “two” in the sense usually used in mathematics in similar contexts: We do not assume that the “two” functions are distinct; the two functions may be different or be the same.
   
   Note on counterexamples: If you need a counterexample, be sure to read the fine print at the beginning. It is great if you can find a counterexample for a general vector space. It is good if you can find a counterexample for a general inner product space. (An “inner product space” is a vector space with an inner product defined on it; an “inner product” is sometimes called a “scalar product”, and it is a generalization of the “dot product” from vector calculus.) It is sufficient if you can find a counterexample for the set of real numbers.  But if you do anything under the assumption that the domain of the functions being considered is NOT a general vector space, be sure to point out explicitly what assumption(s) you are making!
   
   

Last modified Apr 1, 2008 8:40 AM

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