# Re: Function

From: vldm10 <vldm10_at_yahoo.com>
Date: Sun, 20 Jan 2008 11:27:53 -0800 (PST)

On Jan 20, 6:18 am, mAsterdam <mAster..._at_vrijdag.org> wrote:

> This however, IMO complicates without need:
> 'extrinsic'?
> 'sequence of procedures'?
> 'structural programming'?

My intention here was to explain why function is not same as relation, and I separated some important characteristics of the functions in three groups.
There is old definition of the function introduced by Leibniz, and modern definition, defined by the group of mathematicians which wrote under name Bourbaki. The Bourbaki definition became the first to define function in terms of a set of ordered pairs. In Bourbaki definition there is also part which define some additional conditions. Now, there are discussants in this group, who insist that function is just a kind of relation. Although the concept of relation is important, insisting only on this concept can cause a misunderstanding of the function.
In first part I wrote about importance of the rule: "f pairs each x with just one y"
For example almost all laws in physics are in the form of a function, satisfying this simple rule. Very important structures as the sequences are implied by this rule (and not only by this rule). So I don't think that this should go in the glossary, but it should be clarified, because it is about basic things.

> > For presenting function as "f-machine" we can use:
> > Definition1 A function from A to B is a rule that assigns, to each
> > member of set A, exactly one member of set B.
> > Counter - example: In a theatre we can imagine a function from the set
> > of visitors to the set of seats in the theatre. However we can't say
> > what is the rule here. This is weak side of Definition1. We can use
> > set-theoretic Definition2, formally it is OK for this example, but
> > still we don't know what the rule here is. As far as I know there is
> > no good definition for the algorithm; usually the algorithm is defined
> > as the fixed set of rules.
>
> I do not understand what you are saying here.
> Maybe it is a language thing.

This about "f-machine" should be supported by some background and it is Definition1. If you note this definition doesn't have term relation. Definition1 "smells" on a computation process. The counterexample  shows a weak side of Definition1. This can open a discussion about the problems related to a definition of computable or algorithm, what is not necessary for one glossary. Rather this is just one interesting counter-example.

As conclusion, in my opinion you can add Bourbaki definition (Definition2) now or later. So your glossary can cover "computable" approach and also the set abstract approach. It will be OK and enough, although there are some other approaches as the category theory for example.