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Correct use of 3-tuple ?

  1. Aug 4, 2011 #1
    correct use of "3-tuple"?

    (here I use "A" for the universal quantifier, "E" for the existential quantifier, and "e" to indicate elementhood)

    My present definition for Transitivity of a relation:
    R is a transitive relation on the set B

    AxeB AyeB AzeB [((x,y)eR & (y,z)eR)-->(x,z)eR]

    which I shorten to:
    Ax,y,zeB[((x,y)eR & (y,z)eR)-->(x,z)eR]

    But for a certain proof I need Ey rather than Ay, so I'm wondering if I can use a 3-tuple, (x,y,z), in the following way:

    A(x,y,z)eB[((x,y)eR & (y,z)eR)-->(x,z)eR]

    or would that mean I'd have to be using B^3??
  2. jcsd
  3. Aug 13, 2011 #2
    Re: correct use of "3-tuple"?

    Not sure what you're asking, but transitivity is a property of R, not an operation on the x,y, and z. Thus you can't use this definition to establish the existence of y, or for that matter, x, z, or R itself.

    A 3-tuple would be over [tex]B^3[/tex]
    Last edited: Aug 13, 2011
  4. Aug 13, 2011 #3
    Re: correct use of "3-tuple"?

    Hello, EdgeOfWorld (famous last words, there!), yes, I think your original statement,

    [tex](\forall x,y,z \in B) [(((x,y)\in R) \& ((y,z) \in R)) \Rightarrow ((x,z) \in R)],[/tex]

    is equivalent to

    [tex](\forall(x,y,z)\in B^3)[(((x,y)\in R) \& ((y,z)\in R))\Rightarrow ((x,z)\in R)].[/tex]

    I don't see how this relates to the existence of y, but maybe that's because you haven't told us exactly how you're going to use it.
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