# Math 301 (Spring 2024)

(Last updated Wednesday, Feb. 28 @ 4pm)

## Course Information

Instructor: Professor Nicholas Vlamis

Instructor Office: 507 Kiely

Instructor Email: nicholas.vlamis@qc.cuny.edu

Class Meeting: Monday/Wednesday 1:40–3:30pm in Kiely 320

Office Hour: Monday 4–5pm and Wednesday 12–1pm (or by appointment)

Contact: You can always reach me via email or on Discord. I will try to respond within 24 hours.

## Exam Dates

Exam 1: Wednesday, March 6 in class

Exam 2: Wednesday, April 10 in class

Exam 3: Wednesday, May 22 1:45–3:45pm in Kiely 320

## Materials

A place to ask questions and discuss course content (including homework). An online version of LaTex, which is a mark-up language for writing mathematics (or any document once you become a convert). You can also run LaTex natively on your computer. I'm a Mac user, so I can point you to MacTex, but there are also options for Windows and Linux. I really enjoyed writing homework in LaTeX when I was a student, and I really do think it made me a better student. ## Homework Assignments

HW1 (TeX), HW2 (TeX), HW3 (TeX), HW4 (TeX), HW5 (TeX)

## Solutions to Graded Homework Problems

## Week 5 (Week of February 26)

Suggested reading: Section 3.3, Section 4.1

Exam 1 is on Wednesday, March 6. The exam will cover the course material through Week 4, which includes HW1, HW2, HW3, and HW4. The majority of exam questions will be directly taken from or very closed related to problems given on the homework assignments (all problems, not just the starred ones). You may bring a sheet of notes to the exams that includes only definitions and theorem/proposition statements from class and the textbook (up through Section 3.2).

Homework Assignment #5. Due Wednesday, March 13.

Monday's class: Introduced the notion of a subgroup and saw a plethora of examples.

Wednesday's class: Introduced the notion of a cyclic group and cyclic subgroups.

## Week 4 (Week of February 19)

Suggested reading: Section 3.2

Note: We have class on Thursday, February 22, as CUNY will follow a Monday Schedule.

Homework Assignment #4. Due Wednesday, Februrary 28.

Monday's class: Introduced the notion of a group and discussed basic examples.

Wednesday's class: Introduced a more extensive list of exampls of groups, and we established some basic properties of groups. Our class was interrupted by a campus emergency, and so please make sure read Section 3.2. In particular, we were in the process of finishing the proof of Proposition 3.21 when class was interrupted. I was going to finish the lecture portion of the class with stating an immediate corollary of Prop 3.21, which is Prop. 3.22 in the book.

## Week 3 (Week of February 12)

Suggested reading: Section 3.1

Homework Assignment #3. Due Thursday, Februrary 22.

Note: We have class on Thursday, February 22, as CUNY will follow a Monday Schedule.

No class Monday.

Wednesday's class: Introduce equivalence modulo n. Established basic properties of modular arithmetic.

## Week 2 (Week of February 5)

Suggested reading: Section 2.1 and Section 2.2

Homework Assignment #2. Due Wednesday, Februrary 14.

Monday's class: Introduced the greatest common divisor (gcd) of two integers. Proved that the gcd is a linear combination. Established the Euclidean algorithm for finding the gcd and discussed how to write the gcd as a linear combination using the steps from the Euclidean algorithm.

Wednesday's class: Introduced prime numbers. We proved Euclid's lemma, we proved that every natural number other than one is a product of prime numbers (and stated the Fundamental Theorem of Arithmetic), and we proved there are infinitely many primes. We wrote down the principal of mathematical induction (which was proved on HW1).

## Week 1 (Week of January 29)

Welcome to Math 301! I will update this section with summaries of each of the week's classes plus any additional information.

Suggested reading: We will begin the course by covering Section 2.1 and Section 2.2 in the textbook, but not in the same order as presented in the book. I will take most of Section 1.1 and Section 1.2 as assumed knowledge, which includes the basic language of set and how to work with them. When it comes up, I will briefly discuss the definition of an equivalence relation, so depending on your background, you might find it useful to work through these sections now.

Homework Assignment #1. Due Wednesday, Februrary 7.

Monday's class: Worked on a work sheet to recall the definitions of function, injective, surjective, bijective, and function composition.

Wednesday's class: introduced what it means for one integer to divide another. Proved the division algorithm.