Math E104 Syllabus Fall 2008 hey

8/10/2019 Math E104 Syllabus Fall 2008 hey

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MATH E-104 DISCRETE MATHEMATICS WITH COMPUTER

SCIENCE APPLICATIONS Fall 2008

Instructor: Dr. Thomas Koshy

Telephone: 508-626-4727 (O)

e-mail: [email protected]

1. TEXT: K. H. Rosen, Discrete Mathematics and Its Applications, 6th edition,

McGraw-Hill, New York, 2007.

2. OBJECTIVES:

1. Gaining factual knowledge.

Learning the fundamental concepts, notations, and facts in

discrete mathematics.

2. Learning fundamental principles, theories, or generalizations.

Predicting conclusions via inductive reasoning.

Establishing the validity of such conclusions.

Following and appreciating the development of proofs of theorems.

Applying the course material in problem-solving.

3. Improving rational thinking, decision-making, and problem-solving.

Sharpening problem-solving skills and techniques.

Creating simple and short proofs.

Developing computational and writing skills.

Enhancing the ability for rational thinking.

4. Developing a sense of mathematical maturity.

5. Developing a sense of personal responsibility.

6. Gaining a broad understanding and appreciation of intellectual

pursuit.

7. Raising the intellectual curiosity level.

3. COURSE OUTLINE: Chapters 1-4, 6, and 8

Topics

Discussed:LOGIC:Symbolic logic, fuzzy logic, arguments, and proof techniques.

SETS:Sets, set operations, fuzzy sets, cardinality, recursively

defined sets, and formal languages.

FUNCTIONS AND MATRICES:Functions, special functions, properties of

functions, pigeonhole principle, composition, the summation and product

notations, and matrices.


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ALGORITHMS and THEIR COMPLEXITIES:Standard searching and sorting

algorithms, Division algorithm, divisibility properties, Euclidean

algorithm, nondecimal bases, induction, algorithm correctness, growth of

functions, big-oh, big-omega, and big-theta notations, and complexities

of algorithms.

RECURSION:Recursion, Fibonacci and Lucas numbers, Pell and Pell Lucas

numbers, Pell walks, solving recurrence relations, recursive algorithms,

correctness and complexities of recursive algorithms, and generating

functions.

COMBINATORICS:Fundamental counting principles, permutations,

derangements, combinations, permutations and combinations with

repetitions, binomial theorem, and the numbers of surjections and

derangements.

GRAPHS:Graph terminology, paraffins and Pascal-like triangles,

isomorphic graphs, paths, cycles, circuits, Eulerian and Hamiltonian

graphs, planar graphs, graph coloring, and Fibonacci trees.

CATALAN NUMBERS: Catalan numbers, permutations revisited, rail road

tracks, and binary and full binary trees.

3. EXAMINATIONS(all closedbook):

Hour Exams: Oct. 14, Nov. 18, Dec. 09 100 points each

Final Exam (cumulative): Tuesday, Jan. 13, 2009 200 points

4. HOMEWORK: 100 points

Homework problemsare assigned every week and are due every Tuesday.

They are due in class and no late arrivals are accepted, except in case

of emergency or death in immediate family (see the para on make-ups).

They are worth 100 points.

All assignments must be done independently without any outside help; in

other words, no collaborative work is acceptable. Solutions must be

complete with complete justifications for full credit. Solutions that

look similar will be assigned zero grade.

5. PROGRAMMING ASSIGNMENTS: 100 points

Mandatory for graduate students.

Penalized at 5% per weekday for late arrivals.

Not accepted two weeks after due dates.

Last day for submitting assignments is January 06, 2009

6. MAKEUPS:

All tests must be taken on time. No make-ups will be given except in the

case of a real emergency, serious illness, or death in the immediate

family, which must be substantiated. In such a case, the instructor must


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be contacted as soon as possible. If you fall ill on a test day, a

physician's note to the effect that you would not be able to attend the

class, must be produced. In the case of a death, a copy of an obituary

must be presented. All make-ups, if applicable, must be taken on the

first day you return to the class.

Schedule your appointments to avoid conflicts with the class meetings,

especially, the tests.

7. SCIENTIFIC CALCULATORS:

A scientific calculator is strongly recommenced for use in class and

on examinations. Get used to the various function keys and refer to a

manual as often as needed. If you need to buy one, get a graphing

calculator, say, TI-89; it is a good investment into the future!

8.ATTENDANCE:

Not mandatory, but can be very helpful. If you miss a class, it is your

responsibility to find out from others in class to find out what was

said, discussed, and assigned in class.

9. OFFICE HOURS:

Every Tuesday 5:00-5:30 p.m., after class, or by appointment.

10. TEACHING ASSISTANT:

Rebecca Sikora ([email protected])

11. REVIEW SECTIONS:

TBA

12. SUGGESTED REFERENCES:

K. P. Bogart, Discrete Mathematics, D. C. Heath, 1988.

J. Bradley, Introduction to Discrete Mathematics, Addison-Wesley

1988.

J. A. Dossey, et al, Discrete Mathematics, HarperCollins,

Glenview, Illinois, 1987.

R. L. Graham, et al, Concrete Mathematics, Addison-Wesley, 1990.

R. P. Grimaldi, Discrete and Combinatorial Mathematics: An Applied

Introduction, 5th ed., Pearson, 2004.

R. Johnsonbaugh, Discrete Mathematics, Macmillan, 1986.

K. Kalmanson,An Introduction to Discrete Structures and its

applications, Addison-Wesley, 1986.

B. Kolman, et al,Discrete Mathematical Structures, 4th edition,


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Prentice-Hall, 2000.

T. Koshy, Fibonacci and Lucas Numbers with Applications, Wiley,

New York, 2001.

T. Koshy, Discrete Mathematics with Applications, Elsevier,Burlington, MA, 2004.

T. Koshy, Elementary Number Theory with Applications, 2nd edition,

Academic Press, Boston, 2007.

F. S. Roberts, Applied Combinatorics, Prentice-Hall, Englewood Cliffs,

NJ, 1984.

S. Roman, An Introduction to Discrete Mathematics, 2nd edition,

Harcourt Brace Jovanovich, New York, 1989.

K. A. Ross and C. R. B. Wright, Discrete Structures, 3rd ed.,

Prentice-Hall, Englewood Cliffs, NJ, 1992.

13. GRADING:

Total Number

of Points: Hour-Exams 300 points

Homework 100

Programming Assignments 100

Final Exam (cumulative) 200

Total: 700

Grading Policy:

Your grade for the course depends on the number of points you will have

earned by the end of the course, as defined by the following table:

Range in percentage letter grade 95-100 A 90-94 A- 89-87 B+ 86-83 B 80-82 B- 77-79 C+ 73-76 C 70-72 C- 67-69 D+ 63-66 D 60-62 D- 00-59 E

A Final Word:

Study the material every day; review the material every weekend; do the

assignments in advance; score as many points as possible in each

category. Factors such as class participation could contribute to your

final grade, especially if yours is a borderline case.


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MATH E-104 HOMEWORK Fall 2008

All homework solutions must be legible, complete, and independent, written on regular 8 11 paper. If more

than one sheet is needed, staple all sheets together in an orderly fashion. All assignments are due in class

and no late arrivals will be entertained.

1. Exercises 22, 26c, 26d, 28f, 30f, 38, 56, 58 p. 18

2. Exercises 10d, 26, 32, 50b p. 28

3. Exercises 8, 12, 16, 26, p. 47

4. Exercises 4, 6, 8, 10 p. 58

5. Exercises 2, 8, 10 p. 72

6. Exercises 16e, 18e, 26c, 36, 50 p. 131

7. Exercises 10c, 10f, 14c, 16c, 18d p. 161

8. Exercises 8, 18, 20 p. 191

9. Exercises 16, 26, 32, 34 p. 209

10. Exercises 6, 14, 26, 28 p. 217

11. Exercises 6, 20, 24e p. 229

12. Exercises 6, 16 p. 280

13. Exercises 2d, 4c, 8b, 12, 48d p. 308

14. Exercises 2d, 4d, 6f, 10, 22 p. 456

15. Exercises 4b, 4g, 12 p. 471

16. Exercises 22, 30, 34, 40 p. 345

17. Exercises 12, 18, 22, 28, 32 p. 361

18. Exercises 4, 10, 12, 14, 16a p. 379

19. Exercises 26, 28, 48 p. 609

20. Exercises 2, 8, 18, 24, 36, 38 p. 618

21. Exercises 6, 34 p. 644

22. Exercises 4, 6, 24 p. 665

23. Exercises 6, 10, 18, 20 p. 673


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PROGRAMMING ASSIGNMENTS

Your programs must be your own independent creations. They must have all standard features, such as prologues,

detailed comments, indentation, error-checking, modular, and tabular output with proper headings. They must be as

general as possible and should never abort.

Assignment I Due: Sept. 30

1. Check if a (p q) (q r) (r s) (p s) is a tautology.

2. Determine if (p (q (r s))) (p q) (r s).

Assignment II Due: Oct. 14

3. Read in two sets A and B, where U = {1,2,, n}. Print the bit-representations of A

and B. use them to find the elements in A B, A B, A, A B, A B, and A B,

and their cardinalities.

4. Read in a sequence of characters that include left and right parentheses, each

containing at most 30 characters. Determine if each sequence is consists of legally

paired parentheses.

Assignment III Due: Oct. 28

5. January 1, 2000 fell on a Saturday. Determine the day of the week of January 1,

1776 and January 1, 3000. Print the calendar for February for each year.

6. There are two queens on an 8 8 chessboard. Read in their positions on the

chessboard and determine if one can attack the other.

Assignment IV Due: Nov. 18

7. Suppose you place one grain of wheat on the first square of a 10 10 chess board,

2 on the second square, 4 on the third square, 8 on the fourth square, and so on.

Compute the number of grains of wheat needed for the last square and the total

number of grains on the chess board.

8. Construct a table of values of the function h(n) = n2 2999n + 2248541, where 1460

n 1539, and identify each value as prime or composite.

Assignment V Due: Dec. 02

9. Suppose the Tower of Brahma puzzle consists of n = 20 disks. Print the various

moves and the number of moves needed to transfer them from peg Z to peg Z using peg

Y as an intermediary step.

10. Print all triangular numbers 5000 that are squares, say, (ab)2. Make a table of

the numbers aand a table of the numbers b; Do they follow any patterns? If yes,

predict them.

Assignment VI Due: Dec. 16

11. Print all solutions of the LDE x1+ x2+ x3+ x4+ x5= 11, where xi0, and the

number of such solutions. Redo the problem if xi> 0.

12. Read in a positive integer nand list all subsets and permutations of the set

{1,2,3,..., n} in lexicographic order.


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A FEW SUGGESTIONS

FOR

STUDYING MATHEMATICS

1. Read a few sections in advance before each class.

2. Always go to the class prepared. Be prepared to ask and answer questions.

3. Do not skip classes. The information is cumulative; every step counts.

4. Always take down notes (on the RHS of your notebook; use the LHS for problems

in the book).

5. Always have some scrap paper and a pencil with you.

6. Study the material taught in class on the sameday.

7. Write the definitions, properties, and theorems in your own words. Develop a working

vocabulary on the subject. (Remember, math is a language.)8. Keep on writing formulas, definitions, and facts, preferably in your own words.

9. Study the examples done in class; close your notebook; try to do them on your own.

If you cannot do them without any help, study them again and then try again.

Similarly, study the examples in the text and do them yourself.

10. After studying (as in step 9) the relevant portion of the section, do the

corresponding exercises at the end of the section.

11. Do not skip steps or write over previous steps.

12. If you can't solve a problem because it involves a new theorem, formula, or some

property, then restudy the relevant portion of the section.

13. Do math every day. Remember, practice is the name of the game.

14. Math can be learned in "small quantities" only.

15. Work with others whenever helpful.

16. Look for help when in doubt or trouble. (e.g. friend or tutor.)

17. Make the best use of your instructor's office hours.

18. Since math is a cumulative subject, you must build a good foundation to do

additional work.

19. Use your time wisely and carefully.

20. Always review earlier materials before each week. Things must be fresh in your mind

to build upon them.


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HOW TO PREPARE FOR AN EXAM?

1. Review the definitions, formulas, and theorems. Practice writing them down.

2. Study every example worked out in class.

3. Do a few typical problems from every section.

4. On the night before the exam, go to bed early and get a good night's sleep;

otherwise you won't be able to concentrate during the exam and your mind will

wander around.

5. Do not stay up late trying to learn any new material.

6. On the morning of the exam, glance over all materials, for easy recollection,

especially formulas and theorems.

HOW TO TAKE AN EXAM?

0. Take enough writing tools (and a good scientific calculator, if allowed for use in

the exam).

1. Take a few minutes to glance over every problem quickly.

2. Next, do the problems that are easiest for you and those that carry the most weight.

3. Always write the formula, if a formula is involved.

4. Save all your work on scrap paper for later verification.

5. Then do the less easy ones.

6. Whenever possible, save enough time for checking your answers at the end.

7. Always save time to double-check the solutions that carry the most weight.

Good Luck

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Math E104 Syllabus Fall 2008 hey