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.
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.
This article serves as the beginning of propositional logic. First, I discuss logical connectives, then constructing truth tables. I also explain tautologies, contradictions, and contingencies. Towards the end, and I consider modus ponens and substitution. This article is a must-read for any wanting to write mathematical proofs.
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.