**This course begins Summer 2021.**

Have you had that one class yet where you fell in love, with math? Introducing Dave’s online math course: Elementary Set Theory. This exceptional course not only gives you an appreciation for all the hard work put into mathematics before our time, but also how open and free it can be.

## Who This Course is For

- Students in college who want to learn more mathematics.
- Anyone interested in elementary set theory.
- Students who are enrolled in Introduction to Proofs and want to improve their grade.
- Anyone interested in majoring in mathematics, physics, or engineering.
- Anyone wanting to learn about set theory, functions, binary relations, and equivalence relations.

## Requirements

There is no required textbook, though you will need an up-to-date web browser, paper, and pen. A calculator is not necessary.

## What Youâ€™ll Learn in Elementary Set Theory

- Introduction to Sets
- Set Operations
- Symmetric Difference
- Cartesian Product and Families
- Finite Unions and Intersections
- Indexed Sets
- Domain and Codomain
- The Image of a Function
- The Preimage of a Function
- Injective Functions
- Surjective Functions
- Composition of Functions
- Inverse Functions
- Binary Relations
- Composition of Relations
- The Image of a Relation
- The Preimage of a Relation
- Reflexive, Symmetric, and Transitive Relations
- Equivalence Relations
- Partitions
- The Fundamental Theorem of Equivalence Relations
- Closures

## Course Description

This course begins with an introduction to sets from a naÃ¯ve point of view, meaning we donâ€™t start with a formal list of axioms. Instead, we focus on writing proofs concerning various set operations such as unions, intersections, power sets, and symmetric difference. Importantly we also explore families of indexed sets, which are crucial in studying subjects like real analysis.

Next, we study functions defined on sets that have no underlying structure. We focus our attention on domain, codomain, and set operations â€”emphasizing writing proofs. Then we focus our attention on one-to-one and onto functions and then characterize when a function has these properties. We also pay special attention to the composition of functions and inverse functions.

After studying sets and functions on sets, we now begin an introduction to binary relations on sets. We discuss the domain and range of a relation and the preimage, composition, and inverse of a relation. From here, we examine reflexive, symmetric, and transitive relations. We focus a great deal of attention on equivalence relations and partitions and then discuss the Fundamental Theorem of Equivalence Relations in great detail.

## Recommended Prerequisites for Elementary Set Theory

I recommend the prerequisite course Logic and Proof. If you’re not sure if this course is for you, checkout the course contents below or find out more by checking out my free articles on introduction to proofs.