Assignment 6

Suggested Problems (not to be turned in)

• Rosen, Section 10.1, ex: 1, 3, 5, 7, 9, 19, 28
• Rosen, Section 10.2, ex: 1, 3, 7, 9, 15, (21, 23, and 25 with explantions), 27, 29

Required Problems, due October 7

Submission Instructions

• Submissions are to be submitted on paper, with one submission from each assigned pair of students, unless a student has explicit (written) permission from the instructor.

• Both partners should collaborate on every problem, and both names should be at the top of the first page. Since this is a collaborative effort, both partners should agree that the work represents their agreement in the correctness of the answers, and the same grade will be recorded for each assignment.

• Problems for each section (each bullet item in the assignment) should begin on a new page, and problems within a section should be in numeric order.

• If a problem requires the writing of a Python program, then the submission should include a printout of the code, a listing of appropriate test runs, and a statement regarding how the test runs provide sufficient evidence to conclude the program works properly.

• Each problem from Rosen's text should be identified with its section number and problem number. Each problem written out as a separate bullet point should be identified in a phrase or sentence, so it is clear which question is being answered.

• When a problem asks for a conclusion, the solution must include both a statement of the conclusion and an argument justifying that conclusion.

• Answers should be placed in Section and problem order, so the first page (more if needed) should give answers for the problems in the first assignment bullet. Answers for subsequent bullets should be in the order of the bullets on the Assignment sheet.

• Although the rules of this course allow the use of sources beyond the discussions of a student pair, any outside references (e.g., from the textbook, from Web sources, from discussions with others, etc.) must be fully cited.

Problems to be Turned In

• Rosen, Section 10.1, ex: 10 (4, 5, 9), 26, 30
• Rosen, Section 10.2, ex: 2, 8, 12, (22 and 24 with explantions), 28, 30

• Consider the proof of Hall's Marriage Theorem, as discussed in Section 10.2 of Rosen's text and in class.
  Write in your own words a paragraph (at least 7 sentences) that describes the ideas behind this proof. That is, you should not focus on the symbols and algebra, but rather on how the various parts of the proof fit together. For example, your paragraph should include answers to these questions.

  • The Theorem contains the phrase "if and only if". How does this phrasing translate into the structure/organization of the proof?
  • The first part of the proof assumes there is a complete matching and shows |N(A)| ≥ |A| for all relevant subsets A. What (in words) is the main argument for this result?
  • The second part of the proof assumes |N(A)| ≥ |A| for all relevant subsets A.
    • Why do you think that mathematical induction is used here rather than a different type of proof?
    • Why is the proof of this part divided into 2 cases?
      • Why not just use case 1 as the entire proof, rather than have a second case?
      • Alternatively, why not just use case 2 as the entire proof, rather than have a separate case 1?
    • To what extent are the proofs of cases 1 and 2 parallel?
    • To what extent are the proofs of cases 1 and 2 different?

created September 28, 2021 revised October 5, 2021
For more information, please contact Henry M. Walker at walker@cs.grinnell.edu.