Chapter 0: sample

Let’s start my first blog of csc236.

In the chapter 0 Preliminaries of Notes (“Notes” below), the author defines “Partial Order” is “if R is antisymmetric and transitive”. At first, I thought the definition is wrong because this definition of this term in the “How to prove it” (“How” below) is “if R is reflexive, transitive, and antisymmetric”. As if the definition in Notes is missing reflexive. From definition 4.4.1 of “How”, R is said to be antisymmetric if x in A and y in A then xRy and yRx implies x = y. Base on that, xRy and yRx are correct if antisymmetric holds. That could to say that antisymmetric includes reflexive, I guess.

The reason of confusion is the sample R3 in the Notes. R3={(a,b): a and b are persons and a is an ancestor of b}. Notes explains that R3 is a partial order. Form definition of reflexive, we can image from R3: I and I are persons and I am ancestor of I. Oh, I become my daddy, or I am my son, so I am I. Does it sound weird? Of course, there is not any defect in the logic. But, I do not think this sample could help us to understand the term. So, this sample is used in the notes is not reasonable.

This is my personal opinion.