# Introduction to Proofs

## Pick-up these articles and videos on Introduction to Proofs so you can learn the invaluable skill of writing rigorous mathematical proofs for all to read.

Introduction to Proofs is a university course designed to prepare an undergraduate student for writing rigorous mathematical proofs. This course is divided into four main topics. The first is Logic and Proof, which covers propositional logic, quantifiers, mathematical proofs, and logical discourse. The second topic is Number Theory, which covers divisibility, mathematical induction, the Euclidean algorithm, and the Fundamental Theorem of Arithmetic. The third main topic is Introduction to Functions, which covers elementary set theory, families of sets, functions, one-to-one and onto functions, and composition and inverse functions. The fourth and final topic is Introduction to Relations, which covers: binary relations, equivalence relations, partial-order relations, well-founded relations, and confluent relations.

## 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.

## Quantifiers and Predicate Logic

Okay, so you know about propositional logic; now comes quantifiers and predicate logic. First, I discuss the universal and existential quantifiers. Then I explain the uniqueness quantifier and negating quantifiers. Towards the end, and I consider counterexamples and combining quantifiers.

## Mathematical Proofs (Using Various Methods)

Are you someone who relies on logic and evidence for solving problems? Mathematical proofs will help you refine and take advantage of this valuable way of thinking as it applies to mathematics and potentially other areas such as philosophy and computer science.

## Logical Discourse Using Rules of Inference

An axiomatic system contains a set of statements dealing with undefined terms and definitions chosen to remain unproven and called axioms. In this article, I cover an example of a simple axiomatic system where the terms point, line, and incidence only have the meaning given by a small collection of axioms. I mean that I carry out a simple logical discourse for incidence geometry involving points, lines, and incidence.