Choice function
Choice function is way to choice a single element from any of sets, belogging to some set of sets.
Definition
Let \(\mathbb S\) be set of such sets that \(\forall S \in \mathbb S: S \ne \varnothing\)
Let \(f : S\in \mathbb S \mapsto x\in S \)
Then \(f\) is called Choice function
Example
If \(\mathbb S\) is set of kinds of goods in a supermarket, (for example, bred, milk, meat and vegetable}. This can be written as follows:
\(\mathbb S= \{ \mathrm{ bred, milk, meat, vegetable}\}\)
Assume, there are no different pieces of bred with the same price.
Assume, there are no different packets of milk with the same price.
Assume, there are no different packs of meat with the same price.
Assume, there are no different vegetables with the same price.
Assume, one poor Mathematician comes to this supermarket with goal to buy:
one peace of any bread,
one packet of any milk,
one pack of any meat and
one piece of any vegetable.
If he chooses the cheapest bread, cheapest milk, cheapest meat and cheapest vegetable, then this determines his "Choice function". In such a case, the Choice function exists.
However, it may happen, that some product is of poor quality, and the Mathematician needs to buy the more expensive piece, but does not know, what is better; then his choice function may become indefinite.
Also it may happen, that two packets of milk have the same price; and also the Mathematician meets the serious problem.
Similar problem is described by the famous philosopher Buridan: his ass also could not choose between two heaps of grass.
In such a way, the choice function may exist, and may not exist; this depends on the kinds of sets that are allowed in their union \(\mathbb S\). The art of a mathematician is to avoid situations when the choice function does not exist.
Existing of the choice function
Some problems and even paradoxes may arise from the assumption that the choice function always exist.
Actually, such an assumption refers to the concept that can be applies to the set of all possible sets of sets. This means that the concept has no limited range of applicability. According to the First of the TORI Axioms, such a concept is not scientific. In particular, in the example above, the Mathematician may have to follow some customs, of some religious (not scientific) concepts for his choice.