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Homework Help: Showing a relation is a partial order on a set

  1. Apr 26, 2012 #1
    1. The problem statement, all variables and given/known data

    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?

    2. Relevant 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].

    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 [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.

  2. jcsd
  3. Apr 26, 2012 #2
    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.
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