## Confluent Relations (using Reduction Relations)

I discuss confluent relations; in particular, we prove Newman’s Lemma: that local confluence, confluence, the Church-Rosser property, and the unique normal forms property are all equivalent for a well-founded relation. After that, I also give a generalization of Newman’s lemma based on the Buchberger-Winkler’s Property.

## Well-Founded Relations (and Well-Founded Induction)

After you learn mathematical induction on the integers, it’s time to understand well-founded induction on sets. In this article, I discuss well-founded recursion as well. After that, I additionally cover descending chains and antisymmetric and irreflexive relations.

## Partial Order Relations (Mappings on Ordered Sets)

In this article, I discuss partial order relations on a set, often known as a partially ordered set or even poset. I work through the proofs of many of the basic properties. After that, I go through several other important topics.

## Equivalence Relations (Properties and Closures)

The reflexive, symmetric, and transitive properties are motivated. After that, I discuss equivalence relations in detail, including partitions. After that, I prove the fundamental theorem of equivalence relations. Then, towards the end, I explain closures. In the end, the reflexive, symmetric, and transitive closures are studied.

## Binary Relations (Types and Properties)

In this article, I discuss binary relations. I first define the composition of two relations and then prove several basic results. After that, I define the inverse of two relations. Then the complement, image, and preimage of binary relations are also covered.

## Composition of Functions and Inverse Functions

In this article, I discuss the composition of functions and inverse functions. I also prove several basic results, including properties dealing with injective and surjective functions. I include the details of all the proofs.

## One-to-One Functions and Onto Functions

In this article, I cover one-to-one functions and onto functions. One-to-one functions are often called injective, and onto functions are called surjective. I worked through the proofs (in detail) of several basic properties for these special types of functions.

## Functions (Their Properties and Importance)

Hasn’t everyone has heard of what a function is? In this article, I define what a function is and discuss the domain and codomain in detail. I also cover the image and preimage of a function. I do not assume anything other than basic elementary set theory.

## Families of Sets (Finite and Arbitrarily Indexed)

In this article, I cover families of sets. I begin by studying Finite Unions and Intersections. After that, I discuss arbitrarily indexed sets. All proofs are completed in detail, and examples are given.

## Set Theory (Basic Theorems with Many Examples)

Have you ever read through the motivating case for elementary set theory? In this article, I discuss elementary set theory basics, including set operations such as unions, intersections, complements, and Cartesian products. Many theorems are proven in detail, and several examples worked through.