Dave4Math / Courses / Elementary Set Theory

Elementary Set Theory

Master elementary set theory. Learn about sets, functions, one-to-one and onto functions, equivalence relations, and partitions. This course takes a beginner through the ins and outs of writing proofs in basic set theory. You will have an excellent understanding of writing formal proofs and know the fundamentals of binary relations.


January 11, 2021

Created by Dave

This course begins Summer 2021.
Current Status
Not Enrolled
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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.


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

What You’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 naïve point of view, meaning we don’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 —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.

Course Content

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David A. Smith

Mathematics Educator

I sincerely believe that the potential in every student can be unlocked. As an accomplished and dedicated instructor, I have firm confidence that I can provide significant value to your studies.

I relish teaching and use it as a channel for my creativity as I seek to deliver real-life and relevant connections between the mathematics that I teach and my students. Promoting an active learning environment and excitement about mathematics is my dream come true.

David Smith (Dave) has a B.S. and M.S. in Mathematics and has enjoyed teaching precalculus, calculus, linear algebra, and number theory at both the junior college and university levels for over 20 years. David is the founder and CEO of Dave4Math.