Course Information:Course: Introduction to Proofs
Professor: Valentin Zakharevich
Office: Krieger 218
Office Hours: By appointment
Reference: An Infinite Descent into Pure Mathematics version 0.4 by Clive Newstead
Lectures: MW 1:30-2:45 in Chemistry Building 112
TA: Kalyani Kansal
TA Office Hours: Monday 3-4 in Krieger 202
Course DescriptionThis course is an introduction to mathematical proofs and the only prerequisite is basic high-school algebra. This course will consist of two parts. The first half of the semester will be taught in the traditional lecture style. In the last 4 weeks of the semester, we will change to the "inquiry-based learning" (IBL) style, where the students develop the material on their own with the help of a carefully created guide. The classroom will also be flipped at this stage: the students, not the instructor, will be presenting the material at the blackboard.
At the end of the day, a proof is an argument which ought to convince the intended audience that a certain statement is true. The person providing the proof and the person reading it should therefore have an agreed upon set of rules that a valid proof should obey. We will start the semester by formally studying a system of such rules, starting with propositional logic. It will quickly become clear that being explicitly aware of logical operations will not only help convince ourselves of validity of a statement, it will also guide us in search of new proofs and in solving problems. At this stage, we will also spend some time learning to read and write mathematical proofs in the style common in mathematical community. With these tools in hand, we will then proceed to the study of sets, which are essential in all fields of mathematics. In the last week of this part of the semester, we will explore computer-assisted proof writing and verification using Coq.
Part II: Inquiry-based learning
In the last 4 weeks of the semester, we will delve a little deeper into a particular subject: the study of metric spaces. During this period, the students will follow a set of scripts provided by the instructor consisting of definitions, examples, lemmas and theorems, with one important feature: the proofs are missing and have to be provided by the students as their homework assignments.
During the class meetings, students will take turns presenting their work at the blackboard while the rest of the class will be tasked with following along and verifying the validity of the statements. Your blackboard presentation is not expected to be completely correct. In fact, both you and your classmates will learn a lot from seeing common mistakes being made and develop skills to detect and avoid them. Every week, each student will meet with the instructor over Zoom to present a subset of the proofs in the scripts.
There will be homework assigned every week with due date information available on the schedule page. You are allowed and encouraged to work on the problem sets with other students, but every student has to submit solutions written in their own words.
It is encouraged, but not required, that you use LaTex for typesetting your homework assignments. LaTex is a powerful markup language for typing up mathematics and is used in many technical fields. Appendix D of our textbook has a nice introduction to the basic features of LaTex. I will make homework files available in LaTex format so you can start using it right away. If this is your first time using LaTex, I recommend that you start by using an online editor (e.g. Overleaf). I included some useful links on the LaTex page
ExamThere will be an exam covering material on metric spaces. You will be asked to reproduce some of the proofs in the scripts or a slight variation of them. As long as you complete the scripts, this should be an easy exam.
Grading PolicyPart I:
Homework - 50%
Part II - Metric spaces:
Submitted scripts - 30%
Zoom presentations - 5%
Slack participation- 5%
Exam - 10%
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