Spaces, Mappings, and Mathematical Structures, with an Introduction to Proof

The main purpose of this course is to introduce students to the world of mathematical formalism, proof writing, and basic proof techniques (contraposition, contradiction, induction, direct proof, and exhaustive case work). This course aids students as they transition into the study of advanced mathematics, where they may encounter abstract mathematical concepts that do not lend themselves to pen-and-paper calculations. Additionally, training in logic and formal proof is helpful outside of mathematics, especially in fields such as theoretical economics, philosophy, law, computer science, and physics. As we learn about proof writing, we draw examples from different mathematical sub-disciplines including abstract algebra (specifically group theory), basic number theory, probability and combinatorics, and basic point-set topology. Because mathematical reasoning relies heavily upon the language of set theory, learning a bit of this language is our first priority.