How can I write down this property of relations

  1. If I have a relation which is not only antisymmetric (##aRb\rightarrow{}\neg{}bRa##) but it also has a property that ##aRb\land{}bRc\rightarrow{}\neg{}cRa##. How can I be sure that this property holds for any string like that? So that ##aRb\land{}bRc\land{}cRd\rightarrow{}\neg{}dRa## without having to write it down forever?

    I though writing down ##aRb\land{}bRc\rightarrow{}\neg{}cRa## was enough, but with just that I can't prove that ##cRd\rightarrow{}\neg{}dRa##. How can I define this property? Does it exist already?
  2. jcsd
  3. micromass

    micromass 20,039
    Staff Emeritus
    Science Advisor
    Education Advisor

    This is called asymmetric, not antisymmetric.

    How do you know that what property holds for any string? You know that asymmetry and the other property hold because you've assumed it holds.

    Do you somehow want to derive the property
    [tex]aRb ~\wedge~bRc~\wedge~cRd~\rightarrow \neg dRa[/tex]
    from the previous two? I think you can easily find examples of relations that are asymmetric and satisfy the second property and such that the above property doesn't hold.

  4. Sorry, I'm having a hard time explaining this.. I want to create a relation for which asymmetry holds as well as this other property that if a relates to b relates to c relates to d and so on, then not only do they not relate the other way due to asymmetry but also that the last element in the "string" does not relate to any of the previous ones (eg d does not relate to b and a, and c does not relate a) which I don't think you can derive from asymmetry.

    My question is: how can I write this property down simply?
  5. LCKurtz

    LCKurtz 8,446
    Homework Helper
    Gold Member

    Perhaps what you are getting at is what is called a partial ordering. Here are a couple of links:
  6. Partial ordering satisfies reflexivity, antisymmetry and transitivity, that is not the relation that I am looking for.

    The relation I am looking for satisfies the following conditions:
    • ¬(a~b) (anti-reflexitivity)
    • (a~b) then ¬(b~a) (asymmetry)
    • if (a~b) and (b~c) then ¬(c~a) (???)
    In the third property, maybe (a~c) but that is not a requirement, the only requirement is that ¬(c~a). Not only that but also if (a~b) and (b~c) and (c~d) then ¬(d~a).
  7. jbriggs444

    jbriggs444 2,219
    Science Advisor

    If you view the relation as a directed graph I think that what you are after is that it be "acyclic".
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