An Infinite Descent into Pure Mathematics was born out of lecture notes I wrote for teaching Concepts of Mathematics, an introductory pure mathematics class at Carnegie Mellon University, in summer 2015.
I wanted to provide my students with a free resource that helped them learn the skills required to do mathematics, such as solving a mathematical problem, defining new mathematical concepts, writing a proof, typesetting mathematics in LaTeX… the list goes on! I was unable to find a resource that emphasised these aspects whilst also covering enough mathematical topics, so I decided to write my own.
The first part of the book covers core material, including basic mathematical reasoning, symbolic logic, sets, functions, induction and relations. These are the concepts that all mathematicians should feel comfortable with. The second part of the book covers basic material in a number of mathematical areas—topics include number theory, combinatorics, real analysis, infinite sets, equivalence relations, probability theory, order theory and structural induction. Throughout, strategies are provided that help readers to see how mathematical concepts and results can be used in a proof.