Introduction to Pure Mathematics for Advanced High School Students consists of a series of lessons in Logic, Set Theory, Abstract Algebra, Number Theory, Real Analysis, Topology, Complex Analysis, and Linear Algebra.
Lesson 3 from this series introduces a few basic algebraic structures. In this lesson we will learn about binary operations, associativity, commutativity, identity elements, invertible elements. We will also look at examples of semigroups, monoids, and groups.
The content in this lesson is mostly self-contained. It does use a bit of notation that was developed in Lesson 2. Specifically, it will be helpful to understand set-builder notation, the membership relation, and the subset relation.
Two PDF files are included with your purchase. The first PDF file consists of notes followed by a problem set. The problems are broken down into five levels of difficulty. The second PDF file consists of solutions to all the problems.
This series is intended to give high school students who are advanced in math an honest and rigorous introduction to pure mathematics. It is unfortunate that most students will not be exposed to real theoretical mathematics until their sophomore year of college and after they have already declared themselves math majors. My hope is that students wishing to explore real theoretical mathematics before committing to becoming a math major will now have the opportunity to do so. And of course I hope that they enjoy experiencing pure mathematics as much as I enjoy writing about it.