Assignment 5

Suggested Problems (not to be turned in)

• Rosen, Section 9.1, ex: 1, 2, 3, 5, 35 (recall ⊕ means exclusive or)
• Rosen, Section 1.1, ex: 7, 15, 31
• Rosen, Section 1.3, ex: 21, 23, 25
• Rosen, Section 1.4, ex: 7, 9, 25
• Rosen, Section 1.7, ex: 7, 17
• Rosen, Section 5.1, ex: 19, 23

Required Problems, due September 23

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 9.1, ex: 4, 42 (recall ⊕ means exclusive or)
• Rosen, Section 1.1, ex: 7, 16
• Rosen, Section 1.3, ex: 25
• Rosen, Section 1.4, ex: 24
• Rosen, Section 1.7, ex: 14
• Rosen, Section 5.1, ex: 22

• Use mathematical induction to prove that, given x > 0, (1 + x)ⁿ ≥ 1 + nx for all n ≥ 0.

• Consider the following questions related to a database involving a course schedule and courses.

  • The following contains the first 3 records in a courseSchedule table with items found in a school's semester schedule.

    +------------------+----------+------+-----------+---------+----------+
    | courseScheduleID | courseID | day  | startTime | endTime | room     |
    +------------------+----------+------+-----------+---------+----------+
    |                1 |        1 | MWF  | 1:50      | 4:00    | Sal 1035 |
    |                3 |        2 | MWF  | 8:00      | 10:10   | Sal 1035 |
    |                5 |        7 | MWF  | 12:40     | 1:40    | Sal 1030 |
      

    Also the following courses table contains information about several hypothetical courses.

    +----------+--------+----------------------------------+---------+
    | courseID | number | title                            | credits |
    +----------+--------+----------------------------------+---------+
    |        1 |    141 | Intro. to Prog.                  |    1.00 |
    |        2 |    199 | PHP, MySQL, Web Applications     |    1.00 |
    |        3 |    241 | Data Structures                  |    1.00 |
    |        5 |    451 | Topics in Computer Science       |    1.00 |
    |        7 |    496 | Senior Seminar in CS II          |    0.50 |
    

  • In the jargon of relations, what are the domains in the courses table.
  • How many items would be retuned by the SQL query (and briefly explain):

      SELECT * FROM courses JOIN courseSchedule;
        

  • How many items would be retuned by the SQL query (and briefly explain):

      SELECT * FROM courses JOIN courseSchedule on courses.courseID = courseSchedule.courseID ;
        

  • Construct a table indicating all items that would be retuned by the SQL query:

      SELECT * FROM courses JOIN courseSchedule on courses.courseID = courseSchedule.courseID where courseSchedule.courseScheduleID = 3;
        

    The table should show all relevant rows, including entries for all relevant fields in the table.

  • Identify a likely primary key for the courseSchedule table (and briefly explain)
  • Would the day and startTime fields constitute a likely secondary key for the courseSchedule table? Explain briefly.

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  Challenging Problem: (for extra credit) The following claims to prove that all ice cream flavors are equally tasty. Since the result is clearly false, identify the specific error in this argument. (For example, at what step does this argument fail and why.)

  Induction Hypothesis: Let P(n) be the predicate if {IC₁, IC₂, IC₃, ..., IC_n} is any set of ice cream flavors. Then each of the flavors in the set are equally tasty.
  
  For conveience in notation, we will use an equal sign (=) to indicate flavors are equally tasty.

  IC₁ = IC₂ = IC₃ = ... = IC_n

  Proof by Induction on n.

  Base case: When n = 1,

  Let {IC₁} be any set with 1 flavor of ice cream. Then, only 1 flavor is represented, so all flavors in the set are qually tasty.

  Inductive case: Assume P(n) and prove P(n+1).

  Set S = {IC1, IC2, IC3, ... ICn, ICn+1} be any set of n+1 flavors of ice cream. Now consider subsets U = {IC1, IC2, IC3, ..., ICn} V = {IC2, IC3, ..., ICn, ICn+1} Since both U and V have n flavors, we can apply the induction hypothesis P(n) to get IC1 = IC2 = IC3 = ... = ICn IC1 = IC2 = ... = ICn = ICn+1 By transitivity—combining the flavors that are equally tasty, we conclude IC1 = IC2 = IC3 = ... = ICn = ICn+1 This proves P(n+1).

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  create July 22, 2021 revised September 15/20, 2021
  For more information, please contact Henry M. Walker at walker@cs.grinnell.edu.