Showing a relation is a partial order on a set

[ironspud]

Homework Statement

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?

Homework 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.

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!

 

[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.