MATH 213 X1, Fall 2002.

Introduction to Discrete Mathematics

The lowest average to receive A- is 90, and the lowest average to receive A is 93.33333333333333333333333333333333333333333333333333333333333333333333...

The deadline for final grades is December 30. They will most likely appear after or about December 23.

Igor Mineyev, 235 Illini Hall.
mineyev@math.uiuc.edu
Office hours. MWF 3:00-4:00 pm.
Text Discrete Mathematics and its Applications, fourth edition, by Kenneth H. Rosen.
Class time MWF 12:00 noon-12:50 pm
Class location 141 Altgeld Hall
Prerequisites MATH 120 Calculus and Analytic geometry, MATH 135 Calculus, or equivalent.

The course is designed to introduce you to mathematical topics that are useful in applied subjects. The topics covered include sets and relations, function, basic counting techniques, recurrence relations, graphs and trees, matrix algebra. There will be emphasis on algorithms, but you will be required to know some proofs as well.

Homework and exams. The homework will be assigned daily and collected weekly. 10-minute quizzes will be given every week. There will be three 50-minute midterm exams and a comprehensive 3-hour final exam on Monday, December 16, 7 pm - 10 pm, in the regular classroom. The time of the final exam will not be changed. Please make you travel arrangements accordingly. The quizzes and exams will not be necessarily the same as in the homework, but rather will be based on the homework and the material presented in class. No textbooks, lecture notes, or calculators are allowed on the exams and quizzes. No late assignments will be accepted. Missed or late assignments, exams, and quizzes count as 0.

Grading policy. Two lowest homework assignments, one lowest quiz and one lowest midterm exam will be dropped. The grading scale will be decided individually for each exam, and all the quizzes and homework, after the scores are obtained, but it is guaranteed that you get at least A- if the score is 90%, at least B- if the score is 80%, at least C- if the score is 70%, and at least D- is the score is 60%. Each of the midterms will count approximately as 17% of the grade, the homework 17%, the quizzes 17%, and the final exam 33%.
The grades will reflect the accomplishment, not the amount of time and effort spent. Though you will need to put time and effort in order to do well in this class. Be prepared to work regularly.

Homework and exams policy. It is important to do the homework after each class, even though it will be collected weekly. Please write the problems in the order they were given. Staple the sheets (no folding over corners). You are expected to write complete, grammatical sentences, and to spell correctly. Working together on homework is encouraged. Copying is unacceptable. Make sure you know the material, otherwise you will not be able to do well on the exam.

There will be no ``extra credit'' or ``make-ups''. The reason one midterm exam, quiz, and two homework assignments are dropped is to allow for personal emergency situations, when there is a strong reason you must miss a class. Do not miss exams and quizzes (and regular classes) for no reason.

Attendance. It is very important to attend the class. If you miss a class, you are still responsible for learning what was discussed. This includes the homework assigned during that class as well.

Tentative schedule.
• Foundations. Highlights of Sections 1.4-1.8
• Mathematical Induction. Section 3.2.
• Counting. Sections 4.1-4.4 and 4.6.
• First exam.
• Recurrences and Inclusion-Exclusion. Sections 5.1-5.3 and 5.5-5.6
• Relations, with an emphasis on equivalence relations. Sections 6.1, 6.3 and 6.5.
• Second exam.
• Graphs. All of Chapter 7. Graphs and multigraphs, isomorphism, Euler walks and paths, Hamiltonians, algorithms, planar graphs, easy coloring.
• Trees. All of Chapter 8. Rooted trees, traversals, binary trees, minimal spanning trees, sorting algorithms. Some proofs will be included.
• Third exam
• Finishing up. Bring your questions for discussion.
• Final exam.

Homework.
• HW1 begins
• 1.4 # 2, 5, 8, 12, 13, 16, 20, 22.
• 1.5 # 4, 15ac, 18, 20, 35, 36, 45, 46.
• 1.6 # 2, 4, 12, 14, 16, 25, memorize the first three formulas on p.76, prove the first two formulas.
• 1.7 # 6, 14, 16, 18, 20, 33.
• HW2 begins
• 1.8 # 1, 3, 7, 21.
• 3.2 # 1, 2, 6-8, 10, 13, 14, 16, 31.
• HW3 begins
• 4.1 # 1, 3, 6-9, 12, 14, 16, 24, 27, 30, 38, 40, 45(use inclusion-exclusion), 48.
• HW4 begins
• 4.2 # 1-9; write proofs of examples 9 and 10, pp.247-248.
• 4.3 # 1, 2, 4, 5abc, 6abc, 7, 11, 12, 14, 16, 20, 55-58; and 22. Prove Pascal's identity in three different ways. (One of them is #58.)
  Often it is useful to use sets. For example, it is a good idea to denote, say,
  A = the set of k-combinations from blah-blah-blah
  And often the problem is to find |A|, so you say that, and then indeed find |A|.
  Make sure to describe the sets you are using.
• HW5 begins.
• 4.4 # 1-2, 6-10, 12-13, 24a.
• 4.6 # 3, 5, 7-8, 11.
• HW6 begins
• 4.6 # 18, 26, 28, 33, 37.
• 5.1 # 3-6, 9-10.
• 5.2 Prove Theorem 2, p.323; # 1, 3defg, 4cdef, 8.
• 5.5 # 1-5, 7, 16.
• HW7 begins
• 5.6 # 12-15; prove Theorem 1, p.364; # 8-11, 16.
• 6.1 # 1, 3-6, 14-18.
• HW8 begins
• 6.3 # 1-7, 10-17.
• 6.5 # 1, 2abce, 3, 5-8; prove Theorem 1 and Theorem 2, pp.411-412.
• HW9 begins (due Friday, Nov.15)
• 6.5 # 11-13, 14ab, 17abce, 18, 25, 26, 29, 34a.
• 7.1 # 1-9.
• 7.2 # 1-11, 13-17.
• HW10 begins
• 7.2 # 27, 29, 30-32.
• 7.3 # 2, 4, 6, 8, 12, 13, 16, 27(13), 35, 36, 41, 42.
• 7.4 # 1, 3-6, 10abc, 11abc, 13; prove Theorem 2, p.472.
• HW11 begins
• 7.5 # 1-16.
• 7.6 # 2-3 (draw each step of Dijkstra's algorithm as on p.495), 4 (draw the final step with all the labels), 5, 14. In 2-4, also draw minimal length trees.
• 7.7 # 1-6, 9.
• HW12 (not to turn)
• 8.1 # 1-3.
• 8.5 # 1-4, 13-14 (draw steps as in Example 3), 16(13-14) (draw steps as in Example 4).

• Always write complete solutions. Correct anwers with no solution do not score points.
• Explicitly specify the answer to each problem.
• Use the equality sign, "=". Quite often, when you need to show that one thing is equal to another, you write a sequence of equalities connecting the two things. This is the best way to present a solution, whenever possible.
• Use the equality sign correctly. The thing on the left side of "=" must actually be equal to the thing on the right side of "=".

  It is wrong to write: 5=Z. The left hand side is a number, and the right hand side is a set (the set of integers). They are not equal.

  It is wrong to write: {a,b,c}-{a,b,d}=c. (Explain why.)

  It is wrong to write: If A={1,2} and B={a,b}, then AxB=(1,a),(1,b),(2,a),(2,b). (Explain why.)

• Do not write statements that are obviously wrong, for example "2=5" or "P(n)=1²+2²+...+n²".(Why?)
  (Or "our teacher is tough". :)
• When you explain things by words, write complete grammatical sentences.
• Write the solutions to the problems in the order they were assigned. Write the homework number, section numbers, then problem numbers.
• Staple the weekly homework together, to make sure no part of it is lost. (No folding over corners.)
• PRINT your "Last Name, First Name" legibly on the top of the homework. Also write "MATH 213 X1".
• Turn the homework at the beginning of the class it is due.

Additional notes.

The second hour exam was on Friday, November 8 (sections 4.6-7.1). The mean was 89.8 and the mean 93.

The third hour exam was be on Monday, December 9. The mean was 91.7 and the median 94. There was homework on Friday, December 6, but no quiz.

Note for the final exam:
Campus police recommends that you never walk alone. If you don't have someone to walk with you when you leave the exam, call 265-7433 for a free ride on the SafeRide service or call 333-1216 for someone to walk you where you need to go.