Advanced Mathematics

Chapter by Chapter (the nitty‐gritty)

Chapter 1: Logic and Proofs

This book is no Principia Mathematica by Alfred North Whitehead and Bertrand Russell, who famously prove 1 + 1 = 2 after 378 arduous pages in their seminal 1910 work on the foundations of mathematics. That said, there are many mathematical proofs in this book, and each and every one of them is intended to act as a learning experience. The first and foremost, of course, being that before anything can be proven true or false, mathematics must be stated in a precise mathematical language, predicate logic. Chapter 1 introduces the reader to sentential and predicate logic and mathematical induction. After two introductory sections of sentential logic and the connectives: “and,” “or,” “not,” “if then, if and only if,” we move up the logical ladder to predicate logic and the universal and existential quantifiers and variables. Sections 1.4 and 1.5 are spent proving theorems in a variety of ways, including direct proofs, proofs by contrapositive, and proofs by contradiction. The chapter ends with the principle of mathematical induction.

Chapter 2: Sets and Counting

Sets are basic to mathematics, so it is natural that after a brief introduction to the language of mathematics, we follow with an introduction to sets. Section 2.1 gives a barebones introduction to sets, including the union, intersection, and complements of a set. Section 2.2 introduces the reader to the idea of families of sets and operations on families ...