Introduction to Advanced Mathematics, MATH 215, Fall 2019


Instructor: Daniel Groves, 727 SEO e.mail

Course webpage: http://www.math.uic.edu/~groves/teaching/2019-20/215/

Syllabus: Download here.

Course hours:

MWF, 12:00-12:50PM, Room 309, Taft Hall.

Office hours:

Mondays 11am, Friday 2pm (in SEO 727).

(Or, you can make an apointment by e.mail or try stopping past my office.)

Text:


Course description:
The goal of this course is to learn how to create and write mathematical proofs, and to learn why one might want to do such a thing. We will introduce and study some important mathematical concepts used in advanced mathematics courses, particularly equivalence relations.

Assessment:
There will be homework for most classes, two in class midterm exams and a final exam. Since there will be a lot of writing, explaining and critiquing in class, there will also be a class participation component of the grade. The relative weighting of these components will be:
  • Homework: 20%
  • Class participation: 10%
  • Midterm exams: 20% each
  • Final exam: 30%

    Exams:

    The first midterm will be on Wednesday, September 25 during the regular class period.

    The second midterm will be on Wednesday, November 6 during the regular class period.

    The final exam will be as scheduled by the UIC registrar.



    Assigned Homework:

  • On Wednesday, August 28, we will start working on this worksheet (Number Theory I). (Updated 9/4/19 to correct Proposition 9.)

  • Due Friday, August 30, at the beginning of class. Prove Proposition 4 and Proposition 6 from Worksheet 1.

  • Due Wednesday, September 4, at the beginning of class. Prove Propositions 7,8 and 9 and Corollary 10 from the first worksheet.

  • In class on Wednesday, September 4, we will probably start working on this worksheet (Number Theory II).

  • Due Friday, September 6, at the beginning of class. Prove Propositions 11, 12 and 13 from the second number theory worksheet.

  • Due Monday, September 9, at the beginning of class. Prove Propositions 14, 17, 18 and 19 from the second number theory worksheet.

  • For class on Wednesday, Septmber 11, come to class ready to prove of present counterexamples to Conjectures 20 and 21. We will start working on this worksheet in class on Wednesday (Number Theorey III).

  • Here are the first six number worksheets in one file, for your convenience. (UPDATED: First six)

  • Due Monday, September 16, at the beginning of class. Prove Proposition 26, Lemma 28 and Theorem 29 from the third number theory worksheet.

  • In class on Monday, September 16, we will probably start working on this worksheet (on Induction).

  • Due Friday, September 20, at the beginning of class. Do this, which will be graded for points.

  • In class in Monday, September 30 we will start working on this worksheet (Sets and Equivalence Relations).

  • Due Wednesday, October 2, at the beginning of class. Give an example of a relation on the integers which is reflexive but not symmetric and not transitive. Give an example of a relation which is not reflexive, not symmetric and not transitive. Give explanations of why these properties do or do not hold in each case.

  • Sometime soon, we will start working on this number theory worksheet (Number Theory IV).

  • Due Monday, October 7, at the beginning of class. Prove Theorem 32 from the third Number Theory Workhseet.

  • Due Wednesday, October 9, at the beginning of class. Prove Theorem 32 from the third Number Theory Worksheet again.

  • Due Friday, October 11, at the beginning of class. Prove Proposition 35 from the fourth Number Theory Worksheet.

  • Due Monday, October 14, at the beginning of class. Do this homework, which will be graded for points.

  • Here is a proof of Theorem 32 from the third number theory worksheet.

  • Due Wednesday, October 16, at the beginning of class. Prove parts 1,2,3 and 4 from Theorem 37 from the number theory worksheets.

  • Due Monday, October 21, at the beginning of class. Do this homework (which will not be graded for points).

  • On Monday, October 21, we will start working on this worksheet (Number Theory V).

  • Here is a worksheet on rational numbers.

  • Due Friday, October 25, at the beginning of class. Do this homework.

  • Due Wednesday, October 30, at the beginning of class. Do items 1, 2, 5 and 7 from Exercise 2 on this worksheet (on quantifiers).

  • Due Friday, November 1, at the beginning of class. Prove Proposition 5 from the Quantifiers Worksheet.

  • Due, Monday November 11, at the beginning of class. Prove Proposition 8 from the Rational Numbers worksheet.

  • Due, Wednesday November 13, at the beginning of class. Prove Corollary 42 and Proposition 43 from the Number Theory V Worksheet.

  • Due, Friday November 15, at the beginning of class. Prove Theorem 44 and Theorem 47 from the Number Theory Worksheet V Worksheet.

  • On Friday November 15, we will start working on this worksheet (Number Theory VI).

  • Due, Monday November 18, at the beginning of class. Prove Theorem 54 from the worksheet Number Theory VI.

  • Due, Wednesday November 20, at the beginning of class. Prove Lemma 55 and Proposition 56 from the worksheet Number Theory VI.

  • Due, Friday November 22, at the beginning of class. Prove Theorem 57 (Euclid's Lemma) from the Worksheet Number Theory VI.

  • On Friday, November 22, we will start working on this worksheet (on Rings).

  • Due Monday, November 25, at the beginning of class. Prove Proposition 4 from the Worksheet on Rings.

  • Due, Wednesday, November 27, at the beginning of class. Prove that Z/nZ satisfies Axiom (A1) of a ring (this is the first part of Proposition 11 on the Rings Worksheet).

  • Here is the last Worksheet (on Fields).

  • Due Friday, December 6, at the beginning of class. Do this homework, which will be graded for points. Here are some solutions for this homework.