Intro to Proofs

Course Overview

A wide gulf exists between what is expected of students in lower-division college mathematics (like the calculus sequence) and upper-division mathematics (like analysis, number theory, group theory, and topology). Students are asked to shift from weekly homework assignments consisting of 10-20 calculation-based homework problems to writing several pages of logically coherent narratives in order to answer 3-5 theoretical mathematical questions. Being able to read and write proofs requires a firm grasp of logic and familiarity with a wide array of common proof techniques. In order to ease the transition between these two very different styles of mathematics, this semester-long class is dedicated to building the skills of reading and writing proofs.

The class will cover the following essential topics: relevant aspects of formal logic (logical connectives, logical equivalence, quantification); common proof techniques (proof by contradiction, contrapositive, induction, exhaustion, counterexample); and basic properties of sets and proving set equivalence. Additional topics include logical paradoxes, basic properties of functions and relations, and the fundamental theorem of algebra. At many points during the term questions of a more philosophical nature may arise. No effort will be made to avoid such issues. This class will also take the opportunity to introduce students to interesting theorems in elementary number theory and analysis, because that’s what makes life worth living.