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

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

    \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 O^t is not a set. May the next option be more correct??

    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
    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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