Math 55: Discrete Mathematics. Spring 2016.

Course Description: Logic, mathematical induction, sets, relations, and functions. Introduction to graphs, elementary number theory, combinatorics, algebraic structures, and discrete probability theory.

Instructor: Nikhil Srivastava, email: firstname at math.obvious.edu

Lectures: TTh 3:30-5:00pm, F295 Haas Auditorium.

Office Hours: Tuesday 5:15-6:15pm and Wednesday 2:30-4:30pm, 1035 Evans Hall.

Course Control Number: 54008.


Announcements


Enrollment Issues

Please contact the registrar or one of the Mathematics undergraduate advisors:

Graduate Student Instructors

Textbook

Discrete Math and its Applications, 7e, Kenneth E. Rosen (UC Berkeley custom edition)

Exams

There will be two in-class midterm exams on Thursday, 2/18 and on Tuesday, 4/5. The final exam will be on Friday 5/13 from 7-10pm. There will be no makeup exams, so please do not plan to travel on these dates. When calculating grades, your lower midterm exam score will be replaced by your final exam score, if it helps. This will allow you to miss one midterm if necessary.

You are not allowed to bring any books, notes, calculators, or scratch paper to any exam.

Quizzes

There will be a weekly quiz in your section on Wednesday, covering the material from the previous week's lecture. When calculating grades the bottom two quizzes will be dropped.

Homework

Homework problems from the textbook will be assigned on the schedule below. All problems assigned in a week will be due in your discussion section the Wednesday of the following week. Late homework will not be accepted. Questions marked with an x are optional.

You are allowed and encouraged to collaborate on the homework, as long as you write up your own solutions and indicate who you worked with. The latter part is absolutely crucial since mathematical writing is one of the main skills you will learn in this course. All problems which say ``show that...'' are asking for a rigorous mathematical proof, and all proofs should be written in complete English sentences.

Complete solutions for each assignment will be posted on this webpage on the due date. One or two problems each week will be graded by the GSIs. When calculating grades, your two lowest homework scores will be dropped.

  1. Homework 1 (problems assigned 1/19 and 1/21) due Wednesday, 1/27. solutions.
  2. Homework 2 (problems assigned 1/26 and 1/28) due Wednesday, 2/3. solutions.
  3. Homework 3 (problems assigned 2/2 and 2/4) due Wednesday, 2/10. solutions.
  4. Homework 4 (problems assigned 2/9 and 2/11) due Wednesday, 2/17. solutions.
  5. Homework 5 (problems assigned 2/16) due Wednesday, 2/24. solutions.
  6. Homework 6 (problems assigned 2/23 and 2/25) due Wednesday, 3/2. solutions.
  7. Homework 7 (problems assigned 3/1 and 3/3) due Wednesday, 3/9. solutions.
  8. Homework 8 (problems assigned 3/8 and 3/10) due Wednesday, 3/16. solutions.
  9. Homework 9 (problems assigned 3/15 and 3/17) due Wednesday, 3/30. solutions
  10. Homework 10 (problems assigned 3/29 ) due Wednesday, 4/6. solutions.
  11. Homework 11 (problems assigned 4/7 ) due Wednesday, 4/13. solutions.
  12. Homework 12 (problems assigned 4/12 and 4/14 ) due Wednesday, 4/20. solutions.
  13. Homework 13 (problems assigned 4/19 and 4/21 ) due Wednesday, 4/27. solutions + graph drawings.
  14. Homework 14 (problems assigned 4/26 ) due Wednesday, 5/4. solutions + graph drawings.

Advice

Do not wait until the last day to do homework. One difference between this course and previous math courses you may have taken is that you will typically not know how to do a problem when you first see it. The process of interpreting a new mathematical statement and constructing a proof establishing its truth or falsehood can be far from straightforward, and is usually less mechanical than the computational problems which you may be more used to. You will often try many things before you find one that works, and you will spend some time being stuck. This is to be expected, and is typical of any creative activity.

Grading

20% Homework and Quizzes, 20% each midterm, 40% Final. The lower midterm score will be replaced by the final exam score, if it helps.

Piazza

This wikipedia-style forum is an excellent place to ask and answer mathematical questions about the material. Please consider posting to Piazza before sending me or the GSIs email. Sign up here. However, do NOT post answers to current or future homework assignments.

Class Schedule

This course covers a lot of material very quickly, and in order to digest it you will have to read the assigned sections before lecture.

DateTopics ReadingsHomework problemsRemarks
T 1/19 Intro, propositional logic § 1.1-1.3 § 1.1: 7, 9bdf, 12bdf, 18, 23, 27
§ 1.2: 5, 16, 18
§ 1.3: 7, 9, 11, 21
Th 1/21 Predicates, quantifiers, rules of inference § 1.4-1.6 § 1.4: 7, 9, 15, 19, 43
§ 1.5: 9ace, 20, 25, 30
§ 1.6: 2, 3bd, 6, 7, 15, 19
T 1/26 Finish rules of inference, Proofs § 1.6, 1.7, 1.8 § 1.7: 1, 8, 12, 16, 18, 24
§ 1.8: 10, 14, 18, 24, 34
Th 1/28 Sets and functions § 2.1-2.3 § 2.1: 10, 18, 22, 26, 42, 46.
§ 2.2: 12, 16de, 18cd, 26bc, 48.
§ 2.3: 1, 4bc, 14, 20.
T 2/2 Finish functions, sequences, cardinality § 2.3, 2.4, 2.5 § 2.3: 28, 34, 40, 44.
§ 2.4: 26ace, 32, 34, 46.
§ 2.5: 2, 10, 20, 40x.
skip recurrences for now
x means optional
Th 2/4 Modular arithmetic, integer representations § 4.1, 4.2 § 4.1: 16, 24, 30, 37, 38, 39.
§ 4.2: 2, 4, 28, 32.
T 2/9 Primes, GCD, Euclidean algorithm . § 4.3 § 4.3: 4aef, 6x, 11, 16ad, 28, 32abc, 36x, 40, 49, 52.
Th 2/11 Inverses, Chinese Remainder Theorem, Fermat's Little Theorem § 4.4 § 4.4: 2, 6bc, 7, 12bc, 16, 17, 20, 29, 30, 38.
T 2/16 Cryptography, RSA, and review § 4.6 § 4.6: 24, 28.
Th 2/18 MIDTERM 1 (in class)
T 2/23 Mathematical Induction § 5.1, 5.2 § 5.1: 4, 10, 19, 36, 49, 54x, 55x, 62x, 64, 72x.
§ 5.2: 4, 9, 12, 14, 36.
x means optional and difficult,
but recommended
Th 2/25 Recursive definitions § 5.3 § 5.3: 4, 6, 8, 12, 14, 17, 20
T 3/1 Counting and the Pigeonhole Principle § 6.1, 6.2 § 6.1: 8, 16, 22a-f, 26, 40.
§ 6.2: 4, 10, 23, 26, 27, 31, 38x.
Th 3/3 Permutations, combinations, binomial coefficients § 6.3, 6.4 § 6.3: 12, 16, 18, 21, 24
§ 6.4: 8, 10, 14, 16, 20, 22.
T 3/8 generalized permutations and combinations § 6.5 § 6.5: 10, 11, 16, 18, 20, 24, 26x, 47. indistinguishable boxes is skipped for now
Th 3/10 Intro to probability § 7.1 § 7.1: 6, 18, 19, 20, 34, 36, 38.
T 3/15 Probability theory, conditional probability, independence § 7.2 § 7.2: 12, 13, 15, 16, 17, 18, 24, 25, 26, 27ac. skip probabilistic method, algorithms
Th 3/17 Bayes' rule § 7.2-7.3 § 7.2: 34
§ 7.3: 6, 10, 12, 16.
§ 7.4: 4,7.
SPRING BREAK
T 3/29 Random variables, expectation § 7.4 § 7.4: 10, 12, 16, 24, 25, 32, 36, 37, 38, 48.
Th 3/31 Variance, review Chapters 1 and 2, 5.1-5.3, 6.1-6.5, 7.1-7.4. practice Midterm 2
T 4/5 MIDTERM 2 (in class)
Th 4/7 Recurrence relations, generating functions 8.1, 8.4 § 8.1: 1, 3, 6x, 8, 12, 29x, 32x
§ 8.2: 17
§ 8.4: 6cd, 8ad, 10b, 32, 34.
T 4/12 Generating Functions, Inclusion-Exclusion 8.4-8.6 § 8.4: 18, 19, 30ace, 39, 41x, 43.
§ 8.5: 14, 22x, 24, 26.
§ 8.6: 2, 8, 14, 16, 18x.
Note: the book starts Fibonacci at 1, not 0.
Th 4/14 Relations, equivalence relations 9.1, 9.5 § 9.1: 6adh, 10, 47x, 49.
§ 9.3: 2b, 4b, 19b, 24, 32.
§ 9.5: 2bcd, 3abc, 10, 16, 18, 22, 46, 49, 68x.
T 4/19 Intro to graphs, examples, bipartite graphs 10.1, 10.2 § 10.1: 11, 13.
§ 10.2: 5, 18, 20abcd (read examples in 10.2), 26, 35abd, 42abc, 44x, 52, 64.
Supplementary notes from CS70.
Th 4/21 More on bipartiteness, connectivity, coloring 10.4 § 10.4: 28, 29, 30, 32, 34, 36, 37x, 38, 43, 45x, 64x. skip 10.3 for now
T 4/26 More coloring, Euler and Hamilton circuits. 10.5 § 10.5: 2, 4, 10
§ 10.8: 17, 18
+ problems not in the book
HW not graded, but will be on the exam.
Th 4/28 Finish Euler circuits, matchings, social networks. no homework
Tu 5/3 (RRR) Review session in class (Nikhil)
Th 5/5 (RRR) Review session in class (Nick and Julian)
Fri 5/13 FINAL EXAM (7:00pm-10:00pm)