Given an IQ test which consists of 3 inputs $I_n$ and 3 outputs $O_n$. From this information you are given a new input $I_4$ and you have to deduce the output $O_4$.
In other words you have to find a function $f$ such that $\forall n: O_n = f(I_n)$ and the solution is $O_4=f(I_4)$
The inputs and outputs may consist of sets of shapes with certain properties and relations. i.e. shapes which themselves belong to sets. e.g. "set of red things." And in turn these properties e.g. "red" belongs to the "set of colours".
The function $f$ changes the properties of the shapes, addds new shapes or deletes shapes.
Is there a mathematical way (using set theory) to describe this problem in terms of set theory?
A typical function $f$ might correspond to "make all the yellow shapes red" or "make all the red square that's above a green square orange" or "put a green square inside all the yellow circles".
But really $f$ is just a function going from one set of things to another.
So if we have all the information about the sets $I_n$ and $O_n$ we should be able to deduce candiates for $f$. (In fact we want some kind of simplest transform.)
One problem I think is that shapes could potentially belong to an infinite number of sets. e.g. we could construct new sets like "the set of things that are next to a red thing an below a blue thing".