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Thread: a question about transitive relation

  1. #1 a question about transitive relation 
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    Why {(1,2)} is a transitive set?


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  3. #2  
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    See if it fits the definition.


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  4. #3  
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    Is (1, 2) supposed to be an atomic ordered pair, or a set containing the numbers 1 and 2?
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    Quote Originally Posted by MagiMaster View Post
    Is (1, 2) supposed to be an atomic ordered pair, or a set containing the numbers 1 and 2?
    This.

    But I'll first assume TS means A={1,2}.
    If you have the set A={1,2}. Then the binary relation is AxA = {(1,1),(2,2),(2,1),(1,2)}.

    Now why is this transitive? Remember that the rule for transitivity basically says, in simple words: ''for all 2 paths I can take, I can take one''.
    So for instance, I can go from 1 to 2. And from 2 to 1. (= 2 paths). That means that I can also go directly from 1 to 1, which is in this case true (1,1). Remember: it needs to be for all 2 paths. Otherwise, it is not a transitive relation.

    You can also say that a relation is transitive if the composition of A on itself is a subset of the relation A.

    ----------------------
    Now if you are saying that you have a subset of a relation {(1,2)}. Then it is not transitive.
    Last edited by AndyDufresne; May 16th, 2014 at 05:23 AM.
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  6. #5  
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    Well, if it's a set containing a single atomic element, then wouldn't it be transitive just because there's no where to go anyway?

    Anyway, transitive sets (Transitive set - Wikipedia, the free encyclopedia) seem to mean something slightly different and it looks like any set of only atomic elements would be a transitive set. (But {{1,2}} wouldn't be.)
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    Hm, it looks like [b]scienceisfun[b] is what we call "hit and run" poster

    Nevertheless, there seems to be soe confusion here (admittedly, by the OP not being precise with his question)

    Assume that the set is an open interval in - that is the topologist's real line with what called the standard topology on - i.e. is the union of all sets of the form

    Recall that the subset contains every real number between 1 and 2 ( there are uncountably many of them btw) EXCEPT 1 and 2 themselves

    Now consider , and obviously, since for the singleton that then.

    Therefore of necessity, hence is a transitive set

    Now consider that the OP's notation referred, as MagiMagister suspected it might (and why wouldn't he) to and element in the set, say, i.e. an ordered pair

    Well, we may consider in this case that is an element in the power set . And from the above may we assume that , and therefore is in ?

    NO.and raises an important concept - given a set whose elements are, say then the power set has as elements which are SUBSETS of and only those, that is for it can never be the case that

    Therefore if and we may have but we may never have which is the important distinction between "belonging" and "including"
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