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Math Help - Partial Order Relation help

  1. #1
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    Partial Order Relation help

    Let I_{a} denote the identity relation on a set a

    Show that if R is a partial order relation on a then R\backslash I_{a} is a strict partial order relation on a

    Show that if S is a strict partial order relation on a then S \cup I_{a} is a partial order relation on a

    I'm pretty sure I have these down I just want to make sure

    For the first one, this one is just straight forward... I mean if you have a partial order relation and then you take out the identity it becomes irreflexive but I'm not sure how to "show" this. Do I just say:

    x \epsilon R \backslash I_{a} means that x cannot be of the form (x,x) making R \backslash I_{a} irreflexive and thus a strict partial order?

    The second is basicly the same but the opposite way.
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  2. #2
    MHF Contributor
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    Often, the easiest and shortest way for both the writer and the reader is to write things down formally.

    Suppose R is a partial order on a. We need to prove that R\setminus I_a is a strict partial order.

    Irreflexivity. Let x\in a. Then (x,x)\in I_a, so (x,x)\notin R\setminus I_a. That was easy.

    Transitivity. Suppose (x,y),(y,z)\in R\setminus I_a, i.e., (x,y),(y,z)\in R, x\ne y and y\ne z. Then (x,z)\in R. It's left to show that x\ne z. Suppose x=z; then by antisymmetry of R we have x = y, a contradiction.

    Thinking back, it is right that we had to use antisymmetry. If every reflexive and transitive (but not necessarily antisymmetric) relation would produce a strict partial order when one subtracts I_a, then it is unlikely that joining a strict order with I_a would produce an antisymmetric relation as the second part of the problem says.

    I recommend similarly writing out the second part in every detail.
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