Re: Function

On 2008-01-15, Marshall wrote:

> Often "function" by itself means "total function" but sometimes it
> doesn't.

The standard reading of "function" implies "total". The only time somebody qualifies it explicitly is when the context includes all these "non-function functions" as well.

> It seems the difference between a total function and a partial
> function is just in what we want to call the domain.

Yes. The concept of partial functions only exists because in the context of relative mathematical properties, natural extensions and restrictions relate partial functions defined on supersets to actual functions on subsets. And of course because functionality is quite a useful and restrictive property of a general relation as well.

> Division over the domain (integer, integer) is partial; division over
> the domain (integer, nonzero integer) is total.

...which is an example of restriction of course.

Really, the weird part about this thread, to me, is how much time is being spent on how various people construct relations, functions and the like. In today's math it's much more common to go with the axiomatic method and simply talk about the properties any such constructs possess. Under that sort of treatment, most of the fuzziness goes away because you can show that the various constructive versions are isomorphic to each other; from the viewpoint of behavior, properties and logic, they are all just models of the same basic mathematical intuition. Such a viewpoint saves you a whole lot of quibbling.

-- 
Sampo Syreeni, aka decoy - mailto:decoy_at_iki.fi, tel:+358-50-5756111
student/math+cs/helsinki university, http://www.iki.fi/~decoy/front
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