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

On Jan 11, 8:54 am, David BL <davi..._at_iinet.net.au> wrote:
> 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).
>
> Yes, overloading "graph" can cause confusion.
>
> 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
>
> http://en.wikipedia.org/wiki/Relation_%28mathematics%29
>
> 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.
>
> Saying that a function is a relation of course makes a lot of sense.
> However there can be some confusion. For example, the co-domain of a
> function can be referred to as one of the domains!

A function is definitely a type of relation (albeit a binary one). A function is defined as (D, C, G) where G is a subset of the cartesian product of DxC, just like all binary relations. However a function is restricted such that a member of D may only appear as the first element of a single ordered pair in G. I have never personally seen any disagreement or confusion over these definitions in mathematics. Received on Fri Jan 11 2008 - 14:17:56 CET