# Re: cdt glossary [Graph] (was: what are keys and surrogates?)

From: David BL <davidbl_at_iinet.net.au>
Date: Fri, 11 Jan 2008 00:54:26 -0800 (PST)

On Jan 11, 5:12 pm, mAsterdam <mAster..._at_vrijdag.org> 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
>
> That is 'graph' meaning 'plot', not 'a collection of vertices and
> edges'. In cdt it is the latter meaning that is mostly used (when
> discussing network and hierarchical databases).

It seems that when you get down to the detailed formalisms different authors have different definitions of relation and function.

I think it makes most sense to consider a function to be the ordered triple (D,C,G) where D is the domain, C the co-domain and G is the graph of the function.

I've always thought of a (mathematical) relation on X1,...,Xk as formally nothing other than a subset of the cartesian product on X1,...,Xk, but I see here

that it could alternatively be defined as the ordered tuple (X1,...,Xk,G) and we refer to X1,...,Xk as the domains of the relation, and G is a subset of the cartesian product on X1,...,Xk and is called the graph of the relation. In that case it is indeed true that formally a function is a relation.

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