Assignment on Proof Review

As you know, the purpose of a formal proof is to provide a logical and convincing argument to the reader concerning the validity of a statement. Altogether, a proof serves as a careful communication between the proof's author and its reader.

Note: This assignment buildings upon the handout, 3 Inductive Proofs and an Alternative Proof for Theorem: n odd implies n2 odd, discussed in class.

Elements of an Effective Proof

Characteristics of well-structured and clear proofs have been discussed in class! See the instructor, should you have questions or comments.

Some key elements include logical structure, clarity, and the target audience.

Logical Structure and Clarity

As with any communication, an argument must be logically structured—the reader should be led through steps piece-by-piece. For example, a proof should start with statements known to be true plus any initial hypotheses that are explicitly asserted. Each subsequent assertion must be justified based on earlier established truths. Throughout, claims should be clearly written, and reasons behind each claim must be included.

In addition, at Sonoma State, academic work is communicated in English (except for foreign language courses). Thus, rules of English grammar, syntax, etc. must be followed. For example, discussions should not contain sentence fragments, pronouns must have proper antecedents, etc.

Target Audience

Further, to be effective, any communication must take into account the background and experience of the person(s) in a designated target audience. Examples:

• if the potential reader has substantial mathematical background (e.g., calculus), then the communication likely can assume some basic understanding of limits and derivatives.
• if the potential reader has background in high school algebra and introductory statistics (but not calculus), then the communication might use topics of means, averages, expected values, standard deviation, and related concepts with little discussion.
• If the potential reader is comfortable with high school algebra (but not calculus or introductory statistics), then the communication might include algebraic manipulations, but arguments involving limits, slopes of graphs, averages, and expected values likely will require moderate discussion.

As another example, within economics, a discussion of marginal revenue and marginal cost simple can identify these functions as the derivatives of revenue and cost functions, if a targeted reader knows calculus. However, a much more descriptive discussion likely would be needed for readers who lack a calculus background.

For CS 454, we will assume students are comfortable with background from the prerequisite courses, particularly CS 242, Discrete Structures, and CS 315, Data Structures, which in turn requires CS 215, Programming II (including basics of Object-oriented Programming), and CS 242, Discrete Structures. However, for CS 454, background in computer architecture, operating systems, or databases cannot be assumed.

All of this suggests that a formal proof should include sufficient detail to guide the reader through the logical steps, but the proof can (and likely should) omit detail that the reader is already comfortable with.

Framework for this Multi-part Assignment

To highlight qualities that yield effective proofs for the CS 454 target audience, consider file proof-samples.pdf, which presents six different proofs of a standard result in set theory that also can be quite useful within computer science:

Theorem: Let S be a set with n elements. Then S has 2ⁿ subsets.

In this assignment, you are to review each proof and consider to what extent the proof is correct, logically organized, clear, and complete for students in the target CS 454 audience.

Notes:

• Each assignment must be written in proper English, using correct grammar and complete sentences. Further, answers must proceed logically down a page. Placing parts of answers in separate blocks scattered around a page is not acceptable. Also, to maintain a logical order, the answer to question 2 must follow the answer to question 1.
• Since questions are reasonably detailed, the narrative for each assignment almost certainly will require at least 6-8 sentences and fill at least half a page.
• Since the assignment already states the questions, there is no need to copy the questions (such a task takes time without adding any value). Just label each answer to make clear what response goes with which question.

Assignment due Thursday, January 25

Consider Proofs 1 and 2, as presented in proof-samples.pdf.

1. For each Proof, answer the following questions—being as specific as you can in each answer. (For example, rather than stating that a part of the proof is not clear, indicate what part(s) need attention and outline how those part(s) could be made clearer.)

   1. To what extent is the proof well structured and clear?
   2. To what extent does the discussion seem complete and convincing?
   3. Is the level of detail in the proof appropriate? (For example: Should some statements be discussed/explained further? Is the presentation wordy, and if so, what might be condensed or omitted?)
   4. What suggestions might you give to improve the proof?

2. Compare Proofs 1 and 2 regarding their effectiveness in presenting a logical argument.

   • Does one proof have a better structure than the other? Explain.
   • Is one proof clearer than the other? If so, in what way(s)? Explain.

Assignment due Tuesday, January 30

1. For each Proof, answer the following questions—being as specific as you can in each answer. (For example, rather than stating that a part of the proof is not clear, indicate what part(s) need attention and outline how those part(s) could be made clearer.)

   1. To what extent is the proof well structured and clear?
   2. To what extent does the discussion seem complete and convincing?
   3. Is the level of detail in the proof appropriate? (For example: Should some statements be discussed/explained further? Is the presentation wordy, and if so, what might be condensed or omitted?)
   4. What suggestions might you give to improve the proof?

2. Comparing Proofs 1, 2, 3 and 4 regarding their effectiveness in presenting a logical argument.

   • Do some proofs have a better structure than the others? Explain.
   • Are some proofs clearer than the others? If so, in what way(s)? Explain.

Assignment due Thursday, February 1

Consider Proofs 5 and 6, as presented in proof-samples.pdf.

1. For each Proof, answer the following questions—being as specific as you can in each answer. (For example, rather than stating that a part of the proof is not clear, indicate what part(s) need attention and outline how those part(s) could be made clearer.)

   1. To what extent is the proof well structured and clear?
   2. To what extent does the discussion seem complete and convincing?
   3. Is the level of detail in the proof appropriate? (For example: Should some statements be discussed/explained further? Is the presentation wordy, and if ao, what might be condensed or omitted?)
   4. What suggestions might you give to improve the proof?

2. Comparing Proofs 1 – 6 regarding their effectiveness in presenting a logical argument.

   • If you had to choose one (or two) of these proofs as being the best, which would you choose and why?
   • If you had to choose one (or two) of these proofs as being the worse, which would you choose and why?

created Fall, 2023 revised Fall, 2023
For more information, please contact Henry M. Walker at walker@cs.grinnell.edu.