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Thread: 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 $\displaystyle t$:

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

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

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

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

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

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

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

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

    and here is when I don't know which is the best way to say the next:
    A transformation function $\displaystyle \delta$ 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 $\displaystyle T^t$ in order to obtain the final tuple at moment $\displaystyle T^t'$:

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

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

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

    Now let us supose that the relationship $\displaystyle drives$ 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:
    $\displaystyle drives^t: P^t \times C^t \rightarrow \{0,1\}$

    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 $\displaystyle \tau$, 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 $\displaystyle person_i$ and $\displaystyle car_i$ means that $\displaystyle drives^t'(person_i,car_i)=1$ and a change of deletion means that $\displaystyle drives^t'(person_i,car_i)=0$ ??
    How could I write this??

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

    Thank you.
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