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

Date: Fri, 11 Jan 2008 05:17:56 -0800 (PST)

Message-ID: <cf732335-9c5b-4027-a3d2-4a7246f0117d_at_e25g2000prg.googlegroups.com>

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