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

From: David BL <davidbl_at_iinet.net.au>
Date: Sun, 13 Jan 2008 17:35:29 -0800 (PST)

On Jan 14, 6:57 am, JOG <j..._at_cs.nott.ac.uk> wrote:
> On Jan 13, 8:04 pm, David BL <davi..._at_iinet.net.au> wrote:
>
>
>
>
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> > On Jan 13, 3:48 am, JOG <j..._at_cs.nott.ac.uk> wrote:
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> > > On Jan 12, 8:14 am, David BL <davi..._at_iinet.net.au> wrote:
>
> > > > On Jan 12, 2:24 pm, JOG <j..._at_cs.nott.ac.uk> wrote:
>
> > > > > On Jan 12, 1:05 am, David BL <davi..._at_iinet.net.au> wrote:
> > > > > > 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.
>
> > > > Do you agree that most authors define a binary relation as a set of
> > > > ordered pairs? In an earlier post you said a function is the ordered
> > > > triple (D,C,G). How do you reconcile saying that a function is a
> > > > (binary) relation?
>
> > > Relations are formally described by the ordered triple (D,C,G), but
> > > are often informally described by just G.
>
> > So all those authors that define a binary relation as a set of ordered
> > pairs are being informal? I don't agree with that.
>
> > Check out the section under formal definitions in
>
> > http://en.wikipedia.org/wiki/Relation_%28mathematics%29
>
> > Are you saying that definition 1 is informal?
>
> I don't see why you'd think so - in the article, first the domains,
> x1...xn, are stated. Then the graph is specified as a subset of the
> cartesian product of x1..xn. Seems relatively formal to me - domains
> and a graph (albeit for an n-ary as opposed to a binary relation).

Definition 1 doesn't mention the words "graph" or "domains". Instead it says that a relation L over the sets X1,...,Xk is a subset of their cartesian product. It follows for example (with that formal definition) that all empty relations are equal and in general for a given relation it is not possible to uniquely determine a list of "domains".

I appreciate that you prefer definition 2 (and I agree it's better for developing a "theory of relations"). Received on Mon Jan 14 2008 - 02:35:29 CET

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