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Math Help - Defining a transformation function between sets

  1. #1
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    Defining a transformation function between sets

    Hello,

    I'm writing a work and I do not really know which is the best way to specify the following transformation function. Let me introduce it by a simple example.

    Let us supose that we have defined a tuple of objets and relationships at a given moment t:

    T^t=\langle O^t,drives^t\rangle

    the set of objects is O^t=\{P^t,C^t\}, where:
    P^t=\{person1,person2\}
    C^t=\{car1,car2\}

    and the relationships are drives^t=\{drives(person1,car1),drives(person2,car  2)\}

    Given an initial tuple at moment t, I want to define a transformation function \delta. This function consists in applying the required changes (in terms of addition or deletion) in order to obtain an specific tuple at moment t'. That is, let us supose that the tuple that is wanted to be reached at moment t' is the following:

    T^t'=\langle O^t',drives^t'\rangle, where:

    O^t'=\{person1,person2,person3,car1,car2,car3\}
    drives^t'=\{drives(person1,car3),drives(person2,ca  r2),drives(person3,car1)\}

    Then, in order to obtain the final tuple from the initial tuple, we require to apply a set of changes \tau:

    \tau=\{add(person3),add(car3),delete(drives(person  1,car1)),add(drives(person3,car1)),add(drives(pers  on1,car3))\}

    and here is when I don't know which is the best way to say the next:
    A transformation function \delta defines how a given tuple at a initial moment, can be transformed into another tuple at a final moment, that is, the changes that must be applied to the initial tuple at moment T^t in order to obtain the final tuple at moment T^t':

    \delta: T^t \times \tau \rightarrow T^t'

    But I think this is not correct, may be better in the following way??
    \delta: 2^T^t \times \tau \rightarrow 2^T^t'
    or this one??:
    \delta: 2^T^t \times 2^{\tau} \rightarrow 2^T^t'

    ---------------------------------------------------------------------------

    Now let us supose that the relationship drives consists on a function between a pair of objects such that, if returns 0 means that the person does not drive the car, if returns 1, the person drives the car:
    drives^t: P^t \times C^t \rightarrow \{0,1\}

    If I write this relationship in this way, I could say that a change of addition means that the funcion returns 1 and a change of deletion means that the function returns 0?? How I should specify this??

    May I use the same notation than the first one?? I mean, may I say that exists a set of changes \tau, such that, when applied to the initial tuple we obtain the final one??

    If this is correct, how I could write that a change of addition between a pair of objects person_i and car_i means that drives^t'(person_i,car_i)=1 and a change of deletion means that drives^t'(person_i,car_i)=0 ??
    How could I write this??

    That's all. If someone could help me I would be very grateful

    Thank you.
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