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 :

the set of objects is , where:

and the relationships are

Given an initial tuple at moment , I want to define a transformation function . This function consists in applying the required changes (in terms of addition or deletion) in order to obtain an specific tuple at moment . That is, let us supose that the tuple that is wanted to be reached at moment is the following:

, where:

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

and here is when I don't know which is the best way to say the next:

A transformation function 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 in order to obtain the final tuple at moment :

But I think this is not correct, **may be better in the following way??**

or this one??:

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Now let us supose that the relationship 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:

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 , 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 and means that and a change of deletion means that ??

**How could I write this??**

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

Thank you.