- #1
ironspud
- 10
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Homework Statement
Okay, so here's the problem:
(a) Let [itex]U[/itex] be a universal set and suppose that [itex]X,Y\in U[/itex]. Define a relation,[itex]\leq[/itex], on [itex]U[/itex] by [itex]X\leq Y[/itex] iff [itex]X\subseteq Y[/itex]. Show that this relation is a partial order on [itex]U[/itex].
(b) What problem occurs if we try to define this as a relation on the set of all sets?
Homework Equations
A relation [itex]R[/itex] is a partial ordering if [itex]R[/itex] is a reflexive, antisymmetric, and transitive relation.
A relation [itex]R[/itex] on a set A is reflexive if, for all [itex]x\in A[/itex], [itex]x R x[/itex].
A relation [itex]R[/itex] on a set A is antisymmetric if, for all [itex]x,y\in A[/itex], [itex]x R y\wedge y R x\Rightarrow x=y[/itex].
A relation [itex]R[/itex] on a set A is transitive if, for all [itex]x,y,z\in A[/itex], [itex]x R y\wedge y R z\Rightarrow x R z[/itex].
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 [itex]R[/itex] is reflexive:
Let [itex]a\in X[/itex].
Since [itex]a\in X[/itex], then [itex]a\in X[/itex].
Thus, [itex]X\subseteq X[/itex].
Therefore, [itex](a,a)\in R[/itex].
I think this would make sense if I was trying to prove the relation was a subset of [itex]X\times X[/itex] (right?), but I'm trying to show the relation on [itex]U[/itex] with [itex]X,Y\in U[/itex]. 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!