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.