Confluent Relations using Reduction Relations arrow all pointing together to a single point scaled 1

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 Well Founded Induction in a swirling cloud of numbers down to the ground scaled 1

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 scaled 1

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 a binary network graph being handed off scaled 1

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 graph on a blue background scaled 1

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 to do recipes with various soups ingredients and space for text on wooden background scaled 1

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 using pencil and paper scaled 1

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 busy life with all the daily functions scaled 1

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 venn diagrams scaled 1

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 scaled 1

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.