Re: what are keys and surrogates?

On 14 jan, 03:22, David BL <davi..._at_iinet.net.au> wrote:
> On Jan 14, 7:25 am, mAsterdam <mAster..._at_vrijdag.org> wrote:
>
>
>
> > David BL schreef:
>
> > > Keith H Duggar wrote:
> > >> David BL wrote:
> > >>> Keith H Duggar wrote:
> > >>>> David BL wrote:
> > >>>>> Marshall wrote:
> > >>>>>> An interesting note, by the way:
> > >>>>>> functions are relations ...
> > >>>>> Isn't it more precise to say that the graph of a
> > >>>>> function is a relation?
> > >>>> No, it isn't.
> > >>>>http://mathworld.wolfram.com/Function.html
> > >>> From mathworld a relation
> > >>>    http://mathworld.wolfram.com/Relation.html
> > >>> is defined as a subset of a cartesian product. If a
> > >>> function is a relation why do they define a graph of
> > >>> a function f as
> > >>>     { (x,f(x)) | x in domain of f },
> > >>> as described in
> > >>>    http://mathworld.wolfram.com/FunctionGraph.html
> > [snip]
> > > ...  Wikipedia for example defines "graph of
> > > function" without any such restriction
>
> > >    http://en.wikipedia.org/wiki/Function_%28mathematics%29
>
> > >    http://en.wikipedia.org/wiki/Graph_of_a_function
>
> > [snip]
>
> > Quoted from there:
> > "In mathematics, the graph of a function f is the collection of
> > all ordered pairs (x,f(x)). In particular, graph means the
> > graphical representation of this collection, in the form
> > of a curve or surface, together with axes, etc."
>
> > ISTM this is what I use the word 'plot' for.
>
> > Also from that page (at the start):
> > "For another use of the term "graph" in mathematics,
> > see graph theory".
>
> > In dutch 'grafiek' is a 'plot' (or 'chart'), and
> > 'graph' (another word) is a 'collection of edges and nodes'
> > - maybe it boils down to a homonym problem in english?
>
> I was exposed to the formal notion of a graph of a function in
> university when I studied functional analysis.   Check out the closed
> graph theorem
>
>    http://en.wikipedia.org/wiki/Closed_graph_theorem
>
> This is expressed on Banach spaces and has little to do with any
> visualisation.
>
> > > This however doesn't change the fact that most authors define a
> > > (mathematical) relation as a set of ordered tuples, which means a
> > > function is not a relation (assuming, as most do, that a function has
> > > a defined domain and codomain).
>
> > ?
>
> > How does having a domain and a codomain stops a function from being a
> > kind of relation ? (David Cressey asked a similar question).
>
> Given the graph of a function you can determine its domain and range
> (also called image) but not its codomain.   Therefore a formal
> definition of a function tends to use the triple (D,C,G) where D is
> the domain, C is the codomain and G is the graph of the function.
>
> ISTM most authors only define a mathematical relation as a set of
> tuples (which can be compared to G).

In my experience as somebody who works in a Comp. science and Mathematics department the mathematicians that prefer the (D,C,G) definition of a function, also prefer the definition of relation that makes the domains explicit, and so define a binary relation as a triple (D_1,D_2,G). In that case a function is actually again a special case of a relation. But this is all by no means uncontroversial. If you look at the entry for mathematical relation in Wikipedia you will see that there have been edit wars over this, and even one resulting in a ban.

• Jan Hidders