MTH U165 - Introduction to Mathematical Reasoning

Course Information

Course Description

The main objective of the course is to prepare incoming math majors for more challenging mathematical courses at Northeastern by covering the basics of mathematical reasoning and problem solving. The course focuses on learning to write logically sound mathematical arguments and to analyze such arguments. We will attempt to cover the first five chapters of Scheinerman's book, and as much material as time allows in chapters 7 and 8. Here is a more detailed list of topics to be discussed:

• Chapter 1: fundamentals of a mathematical argument (definitions, theorems, proofs, counterexamples, elements of Boolean Algebra).

• Chapter 2: lists, sets, combinatorial proofs.

• Chapter 3: relations (in particular, equivalence relations), partitions of sets, binomial coefficients.

• Chapter 4: mathematical induction, recurrence relations.

• Chapter 5: functions, the pigeonhole principle, symmetries, permutations.

• Chapter 7: basic number theory (divisibility of integers, modular arithmetic, prime factorizations).

• Chapter 8: basics of the group theory, public key cryptography.

The grading will be based on periodic quizzes and written assignments (40%), the midterm (20%), and the final exam (40%). For the take-home assignments you are encouraged to consult with other students. However, ALL WRITTEN ASSIGNMENTS MUST BE WRITTEN INDIVIDUALLY, IN YOUR OWN WORDS.

Attendance will be taken and you are expected to be present for every class. It is your responsibility to be aware of any changes in the syllabus announced in class. Students are responsible for all information given when they are absent.

If you have a concern about the course or the instructor that cannot be resolved by speaking with the instructor, please contact Professor Alex Martsinkovsky (undergraduate director), 471 LA, x5510, alexmart@neu.edu.

It is University policy that no grade, including an incomplete, can be changed after one year. Exceptions must be authorized by the Academic Standing Committee.

All students without legitimate conflicts will take the final exam at the scheduled time. Do not make travel plans that conflict with the final exam.

Homework:
Sep. 6: read "To the Student" preface (pp. xv-xvii), Sections 1 and 2; exercises: p.xvii, #1,2; p.2, #1.1; pp.5-7, #2.1, 2.2, 2.3, 2.6, 2.7.
Sep. 7: read Section 3; exercises: p.15, #3.1, 3.2, 3.3, 3.6, 3.8.
Sep. 11: read Section 4; p. 25, #4.2, 4.5, 4.8, 4.9, 4.13.
Sep. 13: review all previous topics and problems; p. 25, #4.1, 4.3, 4.4.
Sep. 14: read Section 5; p. 27, #5.1-5.9.
Sep. 18: read Section 6; pp. 32-33, #6.1, 6.3, 6.4, 6.9, 6.10(a,b), 6.11(a,b).
Sep. 20: review sections 5 and 6; pp. 32-33, #6.12, 6.14, 6.16, 6.19.
Sep. 21: read Section 7; pp.43-44, #7.1, 7.4, 7.6, 7.7, 7.9.
Sep. 25: p.45, #7.11, 7.13, 7.14; read Section 8; pp.48-49, #8.1, 8.3, 8.4, 8.5, 8.7.
Sep. 27: review sections 7 and 8; p.81, #2-6.
Sep. 28: read Section 9; pp.57-58, #9.1-9.3, 9.5-9.8.
Oct. 2: read Section 11; p. 74, #11.1, 11.2, 11.5-11.9.
Oct. 4: pp. 74-76, #11.12, 11.14, 11.16, 11.21-11.23.
Oct. 5: read Section 13; pp. 87-89, #13.2, 13.3, 13.6, 13.7, 13.8.
Oct. 11: p. 88, #13.12; read Section 14; p. 96, #14.1.
Oct. 12: pp. 96-97, #14.2, 14.5, 14.7, 14.13; read Section 15 (pp. 98-99); p. 102, #15.1.
Oct. 16: read Section 15 (pp. 100-102); p. 103, #15.2-15.4, 15.7, 15.8, 15.13.
Oct. 18: review sections 14 and 15; p. 103, #15.11-15.12.
Oct. 19: read Section 16; pp. 113-114, #16.2, 16.3, 16.7, 16.8.
Oct. 23: pp. 114-115, #16.10, 16.11, 16.14, 16.15, 16.18, 16.22.
Oct. 25: review the material in sections 1-9 and 11-16, including all quizzes and home-work to date.
Oct. 26: read Section 19.
Oct. 30: pp. 141-142, #19.1-19.4, 19.8; read Section 20.
Nov. 1: p. 142, #19.11; p. 153, #20.1-20.4.
Nov. 2: p. 154, #20.6; read Section 21; p. 168, #21.3(a,b,d), 21.4.
Nov. 6: p. 169, #21.7, 21.8.
Nov. 8: review sections 20 and 21; p. 190, #5-7.
Nov. 9: read Section 22 up to Proposition 22.3; p. 188, #22.1, 22.2 (a-h), 22.16(a-c).
Nov. 13: read Section 22 up to Theorem 22.9; p. 188, #22.2 (i-p).
Nov. 15: review Section 22; p. 191, #13-15, 18(a-c).
Nov. 16: read Section 23; pp. 204-205, #23.2-23.4, 23.9, 23.11, 23.12.
Nov. 20: read Section 24; pp. 210-211, #24.1-24.3, 24.9, 24.10.
Happy Thanksgiving!
Nov. 27: p. 210, #24.4-24.6; read Section 25; pp. 215-216, #25.1(a-c,f-h), 25.9-25.11.
Nov.29: p. 243, #3(a,b), 4-6, 8, 10.
Nov. 30: problems in Section 25 assigned on November 27; p.216, #25.12; p.243, #9.
Dec. 4: review the quizzes and homework.

Quizzes and tests:
Quiz 1: Thursday, Sep. 14 (on Sections 2-4).
Quiz 2: Thursday, Sep. 21 (on Sections 5-6).
Quiz 3: Thursday, Sep. 28 (on Sections 7-8).
There will be no quiz on October 5.
Take-home quiz 1: Thursday, Oct. 12 (on the material covered so far). Due Monday Oct. 16.
Take-home quiz 2: Thursday, Oct. 19 (on Sections 14-15). Due Monday Oct. 23.
Midterm (take-home): Thursday, Oct. 26 (on all the topics covered so far). Due Monday Oct. 30.
Quiz 4: Thursday, Nov. 9 (on Sections 19-21).
Quiz 5: Thursday, Nov. 16 (on Section 22).
Quiz 6: Thursday, Nov. 30 (on Sections 23-25).

I will be available for last minute questions on Thursday December 7 at 10:00 AM - 12:30 PM.

Final Exam: Monday December 11 at 8:00 AM in 12 SL (Snell Library)
(allowed: calculators, one sheet of notes).

GRADES