Re: what are keys and surrogates?

On Jan 15, 9:08 am, Jan Hidders <hidd..._at_gmail.com> wrote:
> On 14 jan, 03:22, David BL <davi..._at_iinet.net.au> wrote:
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> > On Jan 14, 7:25 am, mAsterdam <mAster..._at_vrijdag.org> wrote:
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> > > David BL schreef:
>
> > > > Keith H Duggar 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
> > > [snip]
> > > > ... Wikipedia for example defines "graph of
> > > > function" without any such restriction
>
> > > > http://en.wikipedia.org/wiki/Function_%28mathematics%29
>
> > > > http://en.wikipedia.org/wiki/Graph_of_a_function
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> > > [snip]
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> > > Quoted from there:
> > > "In mathematics, the graph of a function f is the collection of
> > > all ordered pairs (x,f(x)). In particular, graph means the
> > > graphical representation of this collection, in the form
> > > of a curve or surface, together with axes, etc."
>
> > > ISTM this is what I use the word 'plot' for.
>
> > > Also from that page (at the start):
> > > "For another use of the term "graph" in mathematics,
> > > see graph theory".
>
> > > In dutch 'grafiek' is a 'plot' (or 'chart'), and
> > > 'graph' (another word) is a 'collection of edges and nodes'
> > > - maybe it boils down to a homonym problem in english?
>
> > I was exposed to the formal notion of a graph of a function in
> > university when I studied functional analysis. Check out the closed
> > graph theorem
>
> > http://en.wikipedia.org/wiki/Closed_graph_theorem
>
> > This is expressed on Banach spaces and has little to do with any
> > visualisation.
>
> > > > This however doesn't change the fact that most authors define a
> > > > (mathematical) relation as a set of ordered tuples, which means a
> > > > function is not a relation (assuming, as most do, that a function has
> > > > a defined domain and codomain).
>
> > > ?
>
> > > How does having a domain and a codomain stops a function from being a
> > > kind of relation ? (David Cressey asked a similar question).
>
> > Given the graph of a function you can determine its domain and range
> > (also called image) but not its codomain. Therefore a formal
> > definition of a function tends to use the triple (D,C,G) where D is
> > the domain, C is the codomain and G is the graph of the function.
>
> > ISTM most authors only define a mathematical relation as a set of
> > tuples (which can be compared to G).
>
> In my experience as somebody who works in a Comp. science and
> Mathematics department the mathematicians that prefer the (D,C,G)
> definition of a function, also prefer the definition of relation that
> makes the domains explicit, and so define a binary relation as a
> triple (D_1,D_2,G). In that case a function is actually again a
> special case of a relation. But this is all by no means
> uncontroversial. If you look at the entry for mathematical relation in
> Wikipedia you will see that there have been edit wars over this, and
> even one resulting in a ban.

LOL. I agree that in a "single work" one should be consistent one way or the other so that a function is indeed a relation.

It doesn't surprise me that there is controversy over the (D1,D2,G) definition of a relation because it seems unconventional (having many a time seen a relation defined as simply a set of tuples).

In the book "Discrete Mathematical Structures With Applications to Computer Science" Tremblay defines binary relations and functions as sets of ordered pairs, states that a function is a relation, but then subsequently makes use of the codomain on a function. His notation is simple, but is it reasonable? Received on Tue Jan 15 2008 - 02:23:25 CET