Miami University

TOM FARMER

SHORT WRITING ASSIGNMENT FOR STUDENTS

MTH 222T/331T Linear Algebra and Discrete Mathematics

Short Writing Assignment, Due Friday, Week 4.

Goal. Become more familiar with the differences between direct and indirect proofs.

Background. To prove a statement of the form "P implies Q" we have at least these options.

Direct Proof: Assume that statement P is true and show it follows that statement Q is true;

Proof of the Contrapositive: Assume that Q is false and show it follows that P is false;

Proof by Contradiction: Assume that P is true and Q is false--show that a contradiction follows.

In order to illustrate these methods of proof, consider the following result that is proved in our text (Lay) on page 385.

Theorem. If  is an orthogonal set of nonzero vectors in Rn, then S is linearly independent.

Clearly, this theorem has the form "P implies Q", where P is the statement:  is an orthogonal set of nonzero vectors in R, and Q is the statement: S is linearly independent. Your assignment is in two parts.

(Part 1) Offer two other proofs of the same theorem. If the author has presented a direct proof then you should give a proof of the contrapositive and a proof by contradiction. Write your proofs carefully, just as if you were revising the text. That is, use complete sentences and explain your logic.

(Part 2) Consider all three methods of proof and write a short (less than a page) discussion comparing the three. Is one of the methods more difficult to explain than the others in this case? Why do you think the author chose the method he used?

This assignment carries the same weight as a quiz. It will be evaluated both for mathematical accuracy and quality of exposition using the rubric below. I am expecting you to use your word processor and to spend some time revising and editing your work.

Grading Rubric. Mathematical Writing assignments involve mathematical skills and writing skills, so the evaluation criteria include both. The criteria for this assignment are:

• Comprehensiveness (20%)

·all aspects of the assignment are covered

·explanations are thorough

·supporting arguments are included

•  Mathematical correctness (30%)

·appropriate mathematical methods are used

·mathematical steps are correct

·mathematical logic is clear

• Expository quality (40%)

·organization is logical and includes an introduction and conclusion

·transitions from one idea to the next are smooth

·mathematical work is smoothly integrated with the text

·mathematical statements are presented within complete sentences

·grammar and spelling are correct.

• Extras (10%)

·overall impact

·creativity and innovation