I have a question regarding the concept of a partial ordering, specifically with respect to a relation $\mathcal{R}$ defined on a set $A$. Here's the background:
Let $X=\{0,1\}$, and define the relation $\mathcal{R}$ on $A=X \times X$ as follows:
$(a, b) \mathcal{R} (c, d)$ if either of the following conditions holds: i) $a < c$, or ii) $a = c$ and $b \leq d$.
I am asked to demonstrate that $\mathcal{R}$ is a partial ordering. To do so, it must satisfy the properties of reflexivity, antisymmetry, and transitivity.
i) To prove reflexivity, I assume that $(a, b) \in A$ and show that $(a, b) \mathcal{R} (a, b)$.
ii) For antisymmetry, I assume that $(a, b), (c, d) \in A$ such that $(a, b) \mathcal{R} (c, d)$ and $(c, d) \mathcal{R} (a, b)$ are both true. Then, I argue that this implies $a = c$ and $b = d$.
iii) To prove transitivity, I assume that $(a, b), (c, d), (e, f) \in A$ and demonstrate that if $(a, b) \mathcal{R} (c, d)$ and $(c, d) \mathcal{R} (e, f)$, then this implies $(a, b) \mathcal{R} (e, f)$.
While I can follow the logical steps in the proof, I'm having difficulty fully understanding how the values of $a, b, c, d, e,$ and $f$ are related in each case and why certain conclusions are drawn. For instance, when proving antisymmetry, I'm uncertain why the conditions $a = c$ and $b = d$ are necessary and how these are derived from the given conditions.
If someone could provide a more intuitive explanation or a step-by-step breakdown of this proof, I would greatly appreciate it. I want to understand the reasoning behind each step.