Savage A. Mathematical Reasoning & Proofs

Ottawa: University of Ottawa, 2017. - 93 p.These are notes for the course Mathematical Reasoning & Proofs at the University of Ottawa. This is an introduction to rigorous mathematical reasoning and the concept of proof in mathematics. It is a course designed to prepare students for upper-level proof-based mathematics courses. We will discuss the axiomatic method and various methods of proof (proof by contradiction, mathematical induction, etc.). We do this in the context of some fundamental notions in mathematics that are crucial for upper-level mathematics courses.
These include the basics of naive set theory, functions, and relations. We will also discuss, in a precise manner, the integers, integers modulo n, rational numbers, and real numbers. In addition, we will discuss concepts related to the real line, such as completeness, supremum, and the precise definition of a limit. In general, this course is designed to provide students with a solid theoretical foundation upon which to build in subsequent courses.
This course should be of interest to students who feel dissatisfied with mathematics courses in which computation is emphasized over proof and explanation. In MAT 1362, we will rarely perform computations based on standard techniques and algorithms, such as computing derivatives using the chain and product rules. Instead, we will focus on why certain mathematical statements are true and why given techniques work. This is often much more difficult than following an algorithm to perform a computation. However, it is much more rewarding and results in a deeper understanding of the material.