Math 109 - Mathematical Reasoning

Winter 2023

Instructor: Daniel Vallieres

Lectures: MWF 1:00-1:50

Where: PETER 102 WLH 2205

Office hours: MWF 2:30-3:30

Office: AP&M 1220

Email: dhvallieres at ucsd dot edu

Teaching assistant for Section C01: Jacob Keller

Discussion sections: C01 T 8:00-8:50 in AP&M 2301

Office hours: Th 3:30-5:30

Office: HSS 4008

Emails: jjkeller at ucsd dot edu

Teaching assistant for Section C02: Nathan Conlon

Discussion sections: C02 T 9:00-9:50 in AP&M 2301

Office hours: M 5:00-7:00

Office: AP&M 2313

Emails: nconlon at ucsd dot edu

Description: Among all of scientific disciplines, mathematics hold a special place. Starting with Euclid's Elements in 300 B.C., mathematicians developed a rigorous way to communicate mathematical truths to one another based on the axiomatic method and formal logic. As a results, most books in mathematics are terse and hard to read as you probably noticed already by looking at some calculus books. Do you remember the format: Definition - Theorem - Proof, Definition - Theorem - Proof, etc??? And have you ever asked yourself why mathematics books are written this way?

This course will introduce the students to rigorous mathematical proofs and the basic language (naive set theory) used by every mathematician to communicate mathematical results in a precise way. Topics to be covered include: logic and the use of quantifiers, naive set theory, functions, relations and equivalence relations, methods of proof such as direct proof, proof by contradiction, proof by contrapositive, proof by cases, induction, and counterexamples. The goal is to cover parts of the first 22 chapters in your book.

Canvas: All announcements for this course will be made in Canvas. It is your responsibility to check Canvas often or to set up your notifications to your own liking for not missing an announcement. I will also post the grades in Canvas. You can access Canvas by clicking here.

Book: An Introduction to Mathematical Reasoning by Peter J. Eccles

Exams: There will be two midterms given in class:

Wednesday, February 1, 1:00 - 1:50 in WLH 2205

Wednesday, March 1, 1:00 - 1:50 in WLH 2205

I will announce later when and where the final exam will be. No notes, textbooks, calculators are allowed during exams. No make-up exams will be given.

Reading: You will get the most out of this class if you read the book carefully on your own. Every week, I will let you know which chapter in the book you should read.

Homework assignments: There will be weekly homework assignments. You will have to submit your hw assignments to your teaching assistant during your discussion section. You will have to submit your assignment in Gradescope on Tuesday before 8am. The first homework assignment will be due on January 17. Only a sample of the problems you submit will be graded, but you will not know which ones ahead of time.

Help options: I strongly encourage you to go to your discussion sections. This will be your opportunity to ask questions about the lectures, the hw problems that have not been graded or any hw problem you have been struggling with. On the other hand, you are welcome to come to my office hours or to your TAs' office hours to ask any questions about the hw problems prior to the deadline. The TAs and myself will try to give you hints if you are stuck and not sure how to attack one particular problem. I encourage you to work with classmates on homework problems, but you have to submit your own copy that reflects your own understanding.

Grading policy: Homework assignments will be worth 20%, midterm 1 20%, midterm 2 20%, and the final exam 40%. I guarantee that 93% will be an A, 90% will be an A-, 87% a B+, 83% a B, 80% a B-, 77% a C+, 73% a C, 70% a C- and above a 60% a D. I also guarantee that 60% or above is a pass.

In order to encourage you to go to your discussion section, if you go to 80% or more of the discussion sections throughout the quarter, you will get a 5% for free on your final grade for participation, and your final exam will count for only 35% rather than 40% of your final grade.

Accommodations: Students requesting accommodations for this course due to a disability must provide a current Authorization for Accommodation (AFA) letter (paper or electronic) issued by the Office for Students with Disabilities (https://osd.ucsd.edu) Students are required to discuss accommodation arrangements with instructors and OSD liaisons in the department in advance of any exams or assignments

Academic Integrity: Academic integrity is highly valued at UCSD and academic dishonesty is considered a serious offense. Students involved in an academic integrity violation will face an administrative sanction which may include suspension or, in very serious cases, expulsion from the university. Your integrity has great value: Cultivate and protect your academic integrity.

Schedule

This is a tentative schedule and I reserve the right to change it if need be.

Week 1 (01/09):

Monday: We will go over the syllabus, and we will try to motivate the need for rigorous mathematical proofs via Polya's conjecture.

Wednesday: Chapter 1: The language of mathematics.

Friday: Chapter 1: The language of mathematics; Chapter 2: Implications.

Week 2 (01/16):

Monday: Martin Luther King Jr Holiday - No class

Wednesday: Chapter 2: Implications.

Friday: Chapter 3: Proofs

Week 3 (01/23):

Monday: Chapter 4: Proofs by contradiction.

Wednesday: Chapter 4: Proofs by contradiction. Proofs by contrapositive.

Friday: Chapter 5: Proofs by induction.

Week 4 (01/30):

Monday: Chapter 5: Proofs by induction

Wednesday: Midterm 1

Friday: Chapter 5: Proofs by induction

Week 5 (02/06):

Monday: Chapter 6: The language of set theory

Wednesday: Chapter 6: The language of set theory

Friday: Chapter 7: Quantifiers

Week 6 (02/13):

Monday: Chapter 7: Quantifiers

Wednesday: Chapter 7: Quantifiers: Especially the definition of convergence of sequences from calculus

Friday: Chapter 8: Functions, and also relations

Week 7 (02/20):

Monday: Presidents' day Holiday - No class

Wednesday: Chapter 9: Injections, surjections and bijections

Friday: Chapter 9: Injections, surjections and bijections

Week 8 (02/27):

Monday: Chapter 22: Partitions and equivalence relations

Wednesday: Midterm 2

Friday: Chapter 22: Partitions and equivalence relations

Week 9 (03/06):

Monday: Chapter 10: Counting

Wednesday: Chapter 11: Properties of finite sets

Friday: Chapter 12: Counting functions and subsets

Week 10 (03/13):

Monday: Chapter 12: Counting functions and subsets

Wednesday: Chapter 14: Counting infinite sets

Friday: Chapter 14: Counting infinite sets