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CS 103X: Discrete Structures Midterm Exam and Honor Code, Exams of Discrete Mathematics

Stanford UniversityDiscrete Mathematics

The Honor Code of Stanford University and the rules for the CS 103X: Discrete Structures Midterm Exam held on February 8, 2007. The Honor Code outlines the expectations of students and faculty regarding academic integrity. The exam rules include the time limit, materials allowed, and instructions for completing the exam. The exam also includes exercises related to sets, rational and irrational numbers, and 3-dimensional space. The document could be useful as study notes, summaries, or exam preparation for students taking a course on discrete structures or related topics.

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CS 103X: Discrete Structures
Midterm
February 8, 2007
THE STANFORD UNIVERSITY HONOR CODE
ā€¢The Honor Code is an undertaking of the students, individually and collectively:
ā€“that they will not give or receive aid in examinations; that they will not give or receive unper-
mitted aid in class work, in the preparation of reports, or in any other work that is to be used
by the instructor as the basis of grading;
ā€“that they will do their share and take an active part in seeing to it that others as well as
themselves uphold the spirit and letter of the Honor Code.
ā€¢The faculty on its part manifests its confidence in the honor of its students by refraining from
proctoring examinations and from taking unusual and unreasonable precautions to prevent the forms
of dishonesty mentioned above. The faculty will also avoid as far as practicable, academic procedures
that create temptations to violate the Honor Code.
ā€¢While the faculty alone has the right and obligation to set academic requirements, the students and
faculty will work together to create optimal conditions for honorable academic work.
Exams are to be done individually and must represent original workā€”it is a violation of the honor code
to copy or derive exam question solutions from other students or anyone at all, textbooks, or previous
instances of this course.
I acknowledge and accept the honor code:
Signature:
Name (print):
EXAM RULES
ā€¢You have two hours to complete this exam.
ā€¢Do not include your scratch work with your exam. Please work the solutions out on another sheet
of paper and then write your solutions neatly on the exam.
ā€¢You may use Prof. Koltunā€™s lecture notes (Sections 1 through 6) and one double-sided cheat-sheet.
You may not use any other material, such as your own course notes, homeworks, distributed homework
solutions, other sets of lecture notes, books, computers, cell phones, crystal balls, Tarot cards, etc.
ā€¢Write clearly and neatly.
ā€¢Stagger your seats.
GOOD LUCK!
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Download CS 103X: Discrete Structures Midterm Exam and Honor Code and more Exams Discrete Mathematics in PDF only on Docsity!

CS 103X: Discrete Structures

Midterm

February 8, 2007

THE STANFORD UNIVERSITY HONOR CODE

  • The Honor Code is an undertaking of the students, individually and collectively:
    • that they will not give or receive aid in examinations; that they will not give or receive unper- mitted aid in class work, in the preparation of reports, or in any other work that is to be used by the instructor as the basis of grading;
    • that they will do their share and take an active part in seeing to it that others as well as themselves uphold the spirit and letter of the Honor Code.
  • The faculty on its part manifests its confidence in the honor of its students by refraining from proctoring examinations and from taking unusual and unreasonable precautions to prevent the forms of dishonesty mentioned above. The faculty will also avoid as far as practicable, academic procedures that create temptations to violate the Honor Code.
  • While the faculty alone has the right and obligation to set academic requirements, the students and faculty will work together to create optimal conditions for honorable academic work.

Exams are to be done individually and must represent original workā€”it is a violation of the honor code to copy or derive exam question solutions from other students or anyone at all, textbooks, or previous instances of this course.

I acknowledge and accept the honor code:

Signature:

Name (print):

EXAM RULES

  • You have two hours to complete this exam.
  • Do not include your scratch work with your exam. Please work the solutions out on another sheet of paper and then write your solutions neatly on the exam.
  • You may use Prof. Koltunā€™s lecture notes (Sections 1 through 6) and one double-sided cheat-sheet. You may not use any other material, such as your own course notes, homeworks, distributed homework solutions, other sets of lecture notes, books, computers, cell phones, crystal balls, Tarot cards, etc.
  • Write clearly and neatly.
  • Stagger your seats.

GOOD LUCK!

Exercise 1 (20 points). Prove or disprove: For any three sets A, B, C,

(a) C \ (A āˆ© B) = (C \ A) āˆ© (C \ B).

(b) C āˆŖ (A āˆ© B) = (C āˆŖ A) āˆ© (C āˆŖ B).

Exercise 2 (20 points). You have been appointed Postmaster General for a new nation. Your job is to determine what denominations to issue stamps in. Unfortunately, the government wants to be able to charge an amount for each letter or package that corresponds exactly to its weight, so the postage fee could be any integral number of cents. The government is also cheap and wants to reduce costs by only printing a minimal number of types of stamps, and stamps of value less than 5 cents will not be allowed.

(a) What is the minimum number n of integer values v 1 , v 2 ,... , vn (where vi ā‰„ 5 for all i) such that all positive integers can be expressed as linear combinations of them? What is the precise condition that such a minimal set v 1 , v 2 ,... , vn has to satisfy? Prove. Does your answer apply to real life?

(b) Suppose the government caps the lowest possible package charge at c cents, for some c > 50. In a ā€œreal lifeā€ scenario, what is now the minimum number of stamp values v 1 , v 2 ,... (where vi ā‰„ 5 for all i) that you can get away with so as to cover all package weights greater or equal to c? Prove.

Exercise 5 (20 points). Consider n planes in 3-dimensional space so that no two are parallel, any three have exactly one point in common, and no four have a common point. What is the number of 3-dimensional parts into which these planes partition the space? Prove. (You can use the 2-dimensional result without proof.)