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

On Jan 11, 10:17 pm, JOG <j..._at_cs.nott.ac.uk> wrote:
> On Jan 11, 8:54 am, David BL <davi..._at_iinet.net.au> wrote:
>
>
>
>
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> > 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
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> > > > 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
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> > > 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 would word that differently: A function is restricted such that a member of D *shall* appear exactly once as the first element of an ordered pair in G.

Your wording sounds like it's optional, and in fact appears to say elements of D shall not appear in the second element of an ordered pair!

> I have never personally seen
> any disagreement or confusion over these definitions in mathematics.

Really! I have seen a (mathematical) relation formally defined as a subset of a cartesian product (and not an ordered tuple) on many occasions. A quick google gave me the following examples

    http://mathworld.wolfram.com/Relation.html

    http://www.cs.odu.edu/~toida/nerzic/content/relation/definition/definition.html

    http://www.definethat.com/define/6729.htm

    http://www.cs.rutgers.edu/~elgammal/classes/cs205/relations_1.pdf

    http://en.wikipedia.org/wiki/Relation_(mathematics)     [see Definition 1]

    http://web.mat.bham.ac.uk/B.J.Philp/msm1f3-web/Pre_2004/relations.pdf Received on Sat Jan 12 2008 - 02:05:30 CET