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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.

    Thanks!
     
  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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