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Thread: Formalization of tuples

  1. #1
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    Formalization of tuples

    Hello,

    I'm trying to formalize some properties of tuples and I find some problems for relating the elements of the tuple.

    I have the tuple $\displaystyle O$ at time $\displaystyle t$ defined as $\displaystyle O^t=\langle A^t, B^t, C^t, q^t, r^t, s^t\rangle$, where:
    - $\displaystyle A^t$ is a set of elements at time $\displaystyle t$
    - $\displaystyle B^t$ is a set of elements at time $\displaystyle t$
    - $\displaystyle C^t$ is a set of elements at time $\displaystyle t$
    - $\displaystyle q^t: A^t \times B^t \rightarrow \{0,1\}$, where $\displaystyle q^t(a,b)=1$ if $\displaystyle a$ and $\displaystyle b$ are related.
    - $\displaystyle r^t: A^t \times C^t \rightarrow \{0,1\}$, where $\displaystyle r^t(a,c)=1$ if $\displaystyle a$ and $\displaystyle c$ are related.
    - $\displaystyle s^t: B^t \times C^t \rightarrow \{0,1\}$, where $\displaystyle r^t(b,c)=1$ if $\displaystyle b$ and $\displaystyle c$ are related.

    Now, I want to represent that: if at time $\displaystyle t$, there is a $\displaystyle q$ relationship that is 1 between $\displaystyle a$ and $\displaystyle b$, and a $\displaystyle r$ relationship that is 1 between $\displaystyle a$ and $\displaystyle c$, then it must exists a relationship $\displaystyle s$ between $\displaystyle b$ and $\displaystyle c$ that is also 1. I don't really know how to write it. I first tried:

    $\displaystyle \forall q^t(a,b) \in O^t \wedge r^t(a,c) \in O^t \rightarrow s^t(b,c)$

    I think this option is not mathematically correct because $\displaystyle O^t$ is not a set. May the next option be more correct??

    $\displaystyle q^t(a,b)=1 \wedge r^t(a,c)=1 \rightarrow s^t(b,c)=1$

    I don't know how to say it. I would be very grateful If you could help me.

    Thank you.
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  2. #2
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    Quote Originally Posted by klendo View Post
    $\displaystyle q^t(a,b)=1 \wedge r^t(a,c)=1 \rightarrow s^t(b,c)=1$
    This seems correct. You may need to quantify this statement over all a, b, c, and possibly t.

    I'd suggest using relations instead of functions q, r, s (they are characteristic functions of relations).
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  3. #3
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    Thank you for your answer emakarov.
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