# Re: what are keys and surrogates?

From: David BL <davidbl_at_iinet.net.au>
Date: Sun, 13 Jan 2008 18:22:17 -0800 (PST)

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

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).

ISTM most authors only define a mathematical relation as a set of tuples (which can be compared to G).

> > Furthermore, I was correct when I
> > stated that a graph of a function is a relation, according to the more
> > general definition of graph of function, as described in Wikipedia.
>
> Is it more general?
>
> Maybe so.
> Anyway, what is wrong with using 'plot' for this, in order to
> disambiguate - is there some meaning lost?

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