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

On Jan 12, 1:05 am, David BL <davi..._at_iinet.net.au> wrote:
> 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:
>
> > > 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 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.

Bit confused by this - a cartesian product generates a set of ordered tuples (over which a function is a subset), and all the hyperlinks you listed seemed to follow that description. Either way, I was referring specifically to not having seen any disagreement concerning the notion that a function is a relation before.

Anyone that does should of course report it immediately to the mathpolice,  who will no doubt open a can of whoop ass from on high in their lofty thrones made of second-hand schaum's outlines ;)

> A quick google gave me the following examples
>
> http://mathworld.wolfram.com/Relation.html
>
> http://www.cs.odu.edu/~toida/nerzic/content/relation/definition/defin...
>
> 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 - 06:24:11 CET