# Showing a relation is a partial order on a set

1. Apr 26, 2012

### ironspud

1. The problem statement, all variables and given/known data

Okay, so here's the problem:

(a) Let $U$ be a universal set and suppose that $X,Y\in U$. Define a relation,$\leq$, on $U$ by $X\leq Y$ iff $X\subseteq Y$. Show that this relation is a partial order on $U$.

(b) What problem occurs if we try to define this as a relation on the set of all sets?

2. Relevant equations

A relation $R$ is a partial ordering if $R$ is a reflexive, antisymmetric, and transitive relation.

A relation $R$ on a set A is reflexive if, for all $x\in A$, $x R x$.

A relation $R$ on a set A is antisymmetric if, for all $x,y\in A$, $x R y\wedge y R x\Rightarrow x=y$.

A relation $R$ on a set A is transitive if, for all $x,y,z\in A$, $x R y\wedge y R z\Rightarrow x R z$.

3. The attempt at a solution

I'm really lost here. On part (a), I thought I was doing fine at first, but the more I think about it, the more I feel I'm way off base. Here's what I mean:

Proof that $R$ is reflexive:
Let $a\in X$.
Since $a\in X$, then $a\in X$.
Thus, $X\subseteq X$.
Therefore, $(a,a)\in R$.

I think this would make sense if I was trying to prove the relation was a subset of $X\times X$ (right?), but I'm trying to show the relation on $U$ with $X,Y\in U$. With that, I think, being the case, I really have no idea how to proceed.

Anyway, clearly I'm over my head here. If anyone could help me out, I'd really appreciate it.

Thanks!

2. Apr 26, 2012

### ironspud

Nevermind. I just spoke with my professor. Seems I was just over thinking everything.

Still, if anyone would like to provide some insight to part (b), I'd appreciate it.