Math 323 Project IV
Even, Odd, and Cool		 Spring 2008
Due 4:14 PM, Friday, March 28. Final solution should be typed. No rough draft required.
NOT A GROUP PROJECT; each student should turn in a paper.

Part I. (Click on the link for Part I; updated March 8, 8:45 AM; slight correction in Problem 2c, March 24, 7:45 PM.)

Part II. (See additional Notes.) Do the problems below.

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.) 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 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.

  2. Listed are two typical incorrect solutions of Exercise 6.27(h). Explain the relation** between
    1. Ex. 6.27(h) and the “reasoning” used in these incorrect solutions, and
    2. the result of the problem above on even, odd, and cool functions.
    You are supposed to explain the relation** between (a) and (b). In other words, you are to explain how the problem above on even, odd, and cool functions is relevant to the errors listed below which were made in the solutions of Exercise 6.27(h).
    (Hint: “Cool function” means “even and odd function”. “Equivalence relation” means “reflexive, symmetric, and transitive relation”.)
    Incorrect solutions of 6.27(h).
    1. Let  R  and  S  be equivalence relations. Then  R  and  S  are transitive. We know from Exercise 6.27(f) that  R ∪ S ***  is not transitive. Thus,  R ∪ S ***  is not an equivalence relation.

    2. Let  R  and  S  be equivalence relations. Then  R  and  S  are transitive. We know from Exercise 6.27(f) that  R ∪ S ***  is not necessarily transitive. Thus,  R ∪ S ***  is not necessarily an equivalence relation.
    **The word “relation” here is used in the ordinary English sense of the word, not in the mathematical sense of a set of ordered pairs.

    ***In case these symbols do not show up properly in your browser,  “ R ∪ S ”  is  “ R union S ”.

Last modified Mar 24, 2008 7:43 PM


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