Discrete Mathematics with Application Manual (PDF download) by fse11171


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                        SCIENCE APPLICATIONS                                        Fall 2008

Instructor:     Dr. Thomas Koshy
Telephone:      508-626-4727 (O)
e-mail:         tkoshy@frc.mass.edu

 1.   TEXT:    K. H. Rosen, Discrete Mathematics and Its Applications, 6th edition,
               McGraw-Hill, New York, 2007.
               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
               7. Raising the intellectual curiosity level.

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

               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.
                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
                COMBINATORICS: Fundamental counting principles, permutations,
                derangements, combinations, permutations and combinations with
                repetitions, binomial theorem, and the numbers of surjections and
                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 closed book):
                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 problems are 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

                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
              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.

              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!

              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.

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

              Rebecca Sikora (rsikora@fas.harvard.edu)


              ⋅   K. P. Bogart, Discrete Mathematics, D. C. Heath, 1988.
              ⋅   J. Bradley, Introduction to Discrete Mathematics, Addison-Wesley
              ⋅   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,
                      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.
MATH E-104                                   HOMEWORK                                          Fall 2008

All homework solutions must be legible, complete, and independent, written on regular 8½ × 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
                                  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 a and 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 xi ≥ 0, and the
         number of such solutions. Redo the problem if xi > 0.
   12. Read in a positive integer n and list all subsets and permutations of the set
         {1,2,3,..., n} in lexicographic order.
                                                                                  παγε 8

                               A FEW SUGGESTIONS
                             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 same day.
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.
                                                                                     παγε 9

                      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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