# Math Notes

The foundation of just about anything is math, but math in its purest sense is idea without substance. If there were any substance to math, then the math would be applied and not pure.

The notes on these pages represent my interpretation of the work of many individuals, and as many as I can recall are referenced. the proofs contain heavily annotated symbolic notation found in A Logical Approach to Discrete Math by David Gries and Fred Schneider. The book is far from perfect, but the book's thesis that the precision of symbolic logic improves the effectiveness of mathematical proofs is a sound concept.

There is one notable departure from the methods in Gries' and Schneider's book. A Logical Approach ... is somewhat, though not completely, careful about free and bound variables. I will attempt to annotate the relative independence of variables, though I confess to a degree of laziness in this area. Given that there is no development of the concept of free and bound variables herein, the annotations may be briefer than necessary. One may assume that all arbitrary variables are distinct and not otherwise related to each other.

My attempts at translating the proofs in these pages may not always render an accurate result. Any comment or suggestion -- constructive or otherwise -- is appreciated.

The math notes contain many special symbols that may not work on your browser. Current versions of the major browsers seem to display the symbols correctly.

On understanding a concept
I often think that I understand an idea only to discover a deeper meaning. This note describes a practical approach you may use to understand something.

On the Law of the Excluded Middle
The law of the excluded middle is a concept that has come under modern scrutiny with the advent of multi-value logics, but the law of the excluded middle can still be useful in solving modern problems.

On the Liar's Paradox and Russell's Paradox
Two interesting and related paradoxes are the Liar's Paradox and Russell's Paradox.

Set membership in the proof of Theorems
This is a short note comparing the set membership requirements in two different set-related proofs.

Quantifying set operations
Theorems using quantified set expressions
A set operation on a collection of sets may be quantified. The theorems on this page are an example of how a set operation on two sets can be generalized to apply over multiple sets.

Properties of singleton sets
Theorems of singleton sets
A singleton set has special properties because a singleton set contains only one element. The properties of a singleton set can be applied in interesting ways to prove set-related theorems.

Properties of relation images - part 1
Theorems of relation images - part 1
An interesting part of set theory is the theory of relations. Set relations are often described as a way of comparing two members of a set, but relations are also a generalized function. The properties described here apply more to the idea that relations are similar to functions.

The proofs shown here are a based on equational logic systems developed by Dijkstra, Gries, and others. Prior understanding of the methods and theorems are assumed. The references cited on the page are a good source of information.

Properties of relation images - part 2
Theorems of relation images - part 2
The continuation of theorems that describe properties of relation images focusing on the properties of range images and relationships between range and domain.

Properties of relations - part 3
Theorems of relation images - part 3
The concept of subset is often used as a set-model of implication, but set-model of implication can be generalized to include a connecting set relation.

Properties of relations - part 4
Theorems of relation images - part 4
Some relations are also functions. The function property imposes special properties on the range and domain of a relation.

Properties of the Identity Relation
Theorems of the Identity Relation
The Identity Relation is a simple relation that finds itself in many different set-related concepts.

Properties of relation conditions
Theorems of relation conditions
The continuation of theorems that describe properties of relation images focusing on an image expression denoting the condition of a set.

Properties of relation compositions
Theorems of relation compositions
The continuation of theorems that describe properties of relation images focusing on the composition of two relations.

Well-Ordering Principle
The well-ordering principle is fundamental to the theory of integers and the nature of numbers.

Divisibility Concepts
The divisibility of integers has its foundation in the well-ordering principle.