$X - Y$ in a finite set

Question

Question: The domain is the power set of a finite set. $X$ is related to $Y$ if $X - Y$ is not empty. Is it a partial or strict order? If it is a partial order or strict order, is it also a total order?

I don't quite understand this. Can anyone explain it to me in simple terms? Does $X - Y$ mean an element $X$ subtracted by an element $Y$?

Answer 1

The question means the following:

Let $S$ be some set. Let $P$ be the power set of $S$; that is, let $P=\mathcal{P}(S)$. Thus, elements of $P$ are themselves subsets of $S$.

Given any two $X,Y\in P$, we can define their set difference $Y-X\in P$ to be this subset of $S$: $$Y-X=\{s\in S: s\in Y\text{ and }s\notin X\}$$

Define a relation $\mathsf{R}$ on $P$ (in other words, a subset $\mathsf{R}\subseteq P^2$) as follows: given $X,Y\in P$, $$X\;\mathsf{R}\;Y\iff \text{the set }Y-X\text{ is not empty}$$

1. Figure out if $\mathsf{R}$ is a partial order, i.e., whether

   • for all $X\in P$, we have that $X \;\mathsf{R}\; X$
   • for all $X,Y\in P$, we have that if $X \;\mathsf{R}\; Y$ and $Y \;\mathsf{R}\; X$, then $X=Y$
   • for all $X,Y,Z\in P$, we have that if $X \;\mathsf{R}\; Y$ and $Y \;\mathsf{R}\; Z$, then $X \;\mathsf{R}\; Z$

2. Figure out if $\mathsf{R}$ is a strict order, i.e., whether

   • for all $X\in P$, it is false that $X \;\mathsf{R}\; X$
   • for all $X,Y\in P$, we have that if $X \;\mathsf{R}\; Y$, then it is false that $Y \;\mathsf{R}\; X$
   • for all $X,Y,Z\in P$, we have that if $X \;\mathsf{R}\; Y$ and $Y \;\mathsf{R}\; Z$, then $X \;\mathsf{R}\; Z$

3. If $\mathsf{R}$ is either a partial or strict order, figure out if $\mathsf{R}$ is a total order, i.e., whether

   • for all $X,Y\in P$, either $X\;\mathsf{R}\; Y$ or $Y\;\mathsf{R}\; X$