**
Introduction to Advanced Mathematics, MATH 215, Spring 2019
**

**Instructor:** Daniel Groves, 727 SEO e.mail

** Course webpage:**
http://www.math.uic.edu/~groves/teaching/2018-19/215S/

**Syllabus**: Download here.

**Course hours:**

MWF, 2:00-2:50PM, Room 309, Taft Hall.

** Office hours:**

Tuesdays 10am, Fridays 11am (in SEO 727).

**Note:** There will be no office hours on Tuesday, April 30.

(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 **Friday, February 15** during the regular class period.

The ** second midterm** will be on **Friday, April 5** during the regular class period.

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

** Math moves:**

Here is a document containing the `moves' and other advice that I have given in class. I hope to had to it over the course of the semester, with new advice and moves as they come up. Please feel free to e.mail with questions or suggestions for new rules. (UPDATED 1/23)

** Daily Homework:**

On Wednesday, January 16, we will start working on this worksheet in class.

Due **Friday, January 18, at the beginning of class**. Proofs of Propositions 4 and 6 from Worksheet 1.

Due ** Wednesday, January 23, at the beginning of class**. Proofs of Propositions 7,8,9 and 10 from Worksheet 1.

On **Wednesday, January 23** we will probably start working on this worksheet.

Due **Friday, January 25, at the beginning of class**. Proofs of Propositions 11, 12 and 13 from Worksheet 2.

Due ** Monday, January 28, at the beginning of class** Prove Proposition 14 from Worksheet 2, and also give the truth tables for (P ⇒ Q) ∧ R and P ⇒ (Q ∧ R).

On **Wednesday, January 30**, we will probably start working on this worksheet.

Due **Wednesday, January 30, at the beginning of class**, prove Propositions 17, 18 and 19 from Worksheet 2. **UPDATE:** Class cancelled on 1/30, so turn this in on 2/1.

Due ** Friday, February 1, at the beginning of class**. Do this homework, which will be graded for points.

**Due, Monday, February 4, at the beginning of class**. Prove Propositions 23, 24, 25 and 26 from Worksheet 3.

On ** Monday, February 4**, we will take a break from Number Theory and start working on this worksheet.

**Due, Wednesday, February 6**, at the beginning of class. Give three examples of relations on the integers which are not the one I gave in class (r less than s) or any of the ones on the relations worksheet).

**Due, Friday 8**, at the beginning of class. Give two examples of relations on the integers. The first should be reflexive and symmetric but not transitive. The second should be transitive but not reflexive and not symmetric. For each property that holds, explain why it holds. For each property that does not hold, give an example of numbers which show that it does not hold.

** Due, Wednesday, February 13**, at the beginning of class. Prove Lemma 28 from the worksheet 'Elementary Number Theory, III'.

** Due, Wednesday, February 20**, at the beginning of class. Prove Theorem 31 from the third number theory worksheet.

**Here** is the next worksheet, which we will probably start working on on 2/22.

** Due, Friday, February 22**, at the beginning of class. Prove Theorem 32 from the third number theory worksheet.

**Due, Monday, February 25**, at the beginning of class. Prove Corollary 33 from the fourth number theory worksheet.

**Due, Wednesday, February 27,** at the beginning of class. Prove Corollary 33 (again).

**Due, Friday, March 1**, at the beginning of class. Do this homework, which will be graded for points.

**Due, Monday March 4** at the beginning of class. Do the first four exercises from class on Friday: **Z/nZ** is the set of equivalence classes of integers mod n, with addition defined by [a]+[b] = [a+b] and multiplication by [a][b] = [ab]. Define a zero for **Z/nZ** and prove that with this zero you have [a] + 0 = [a]; define a one, and prove that 1.[a] = [a]. (Note that both the 0 and the 1 are your choice of equivalence classes, not numbers. In class we denoted them with subscripts.) Then prove that for all classes [a] and [b] we have [a]+[b] = [b]+[a] and [a][b] = [b][a].

**Due, Wednesday, March 6** at the beginning of class. Prove Proposition 35 from the fourth number theory worksheet.

**Due, Friday, March 8** at the beginning of class. Prove statements 1,2,3 and 4 from Proposition 37 from the fourth number theory worksheet.

**Due, Monday, March 11**, at the beginning of class. Prove Theorem 39 and Corollary 40.

Here are the first **six** number theory worksheets in one file.

On Monday, March 11, we will start working on this worksheet.

**Due, Friday, March 15**, at the beginning of class. Prove Propositions 5, 6 and 7 from the Rings and Fields Worksheet.

**Due, Wednesday, March 20**, at the beginning of class. Do this homework, which will be graded for points.

**Here** is the next number theory worksheet. (It has also been included in the single PDF document linked above.)

**Due, Friday, March 22**, at the beginning of class. Prove Theorem 44 and Proposition 45.

On **Monday, April 8**, we will probably start working on this worksheet. (It has also been placed into the single document with all of the number theory worksheets.)

**Due Wednesday April 10, at the beginning of class.** Prove Theorem 55 from the sixth number theory worksheet.

**Due Friday April 12, at the beginning of class.** Prove Theorem 55 again.

**Due Monday April 15, at the beginning of class.** Prove Lemma 56, Proposition 57 and Theorem 58.

Starting ** Monday April 15**, we will start working on these worksheets: Fields and Induction.

**Due, Wednesday April 17, at the beginning of class.** Prove Proposition 5 from the Fields worksheet. You may assume that you already know **Z**/p**Z** is a ring, so you just need to prove that Axiom (M4) holds.

** Due Monday, April 22, at the beginning of class.** Do Exercise 3 from the Induction Worksheet.

**NOTE:** There will be no office hours on Tuesday, April 30.

** Due Wednesday, April 24, at the beginning of class.** Prove Proposition 10 from the Induction Worksheet.

On Friday, April 26, we will work on this worksheet (on Quantifiers and Negation).

**Due Monday, April 29, at the beginning of class.** Prove that the statement in Section 0.5 `Disproving statements with quantifiers' on the Quantifiers and Negation worksheet is indeed false. (NOTE: For the 7 people missing from class on Friday April 26, we spent most of the class working on this problem, so you may want to talk to classmates who were there.)

Due, **Wednesday May 1, at the beginning of class**. Do this homework, which will be graded for points.