Re: I think my book may be wrong about cardinality, but I'm not sure

From: Cimode <cimode_at_hotmail.com>
Date: Thu, 26 Jul 2007 14:24:57 -0700

On 26 juil, 22:53, "Bruce C. Baker" <b..._at_undisclosedlocation.net> wrote:
>
> news:46a8fca4\$0\$8833\$9a566e8b_at_news.aliant.net...
>
>
>
> > Bruce C. Baker wrote:
>
> >> "Cimode" <cim..._at_hotmail.com> wrote in message
>
> >>>On 26 juil, 18:21, "Bruce C. Baker" <b..._at_undisclosedlocation.net>
> >>>wrote:
>
> >>>>"Cimode" <cim..._at_hotmail.com> wrote in message
>
>
> >>>>>On 26 juil, 16:38, "Bruce C. Baker" <b..._at_undisclosedlocation.net>
> >>>>>wrote:
>
> >>>>>>"Cimode" <cim..._at_hotmail.com> wrote in message
>
>
> >>>>>>>On 26 juil, 03:13, "Bruce C. Baker" <b..._at_undisclosedlocation.net>
> >>>>>>>wrote:
>
> >>>>>><snip>
>
> >>>>>>>>Each tuple in a relation with N attributes corresponds to a point in
> >>>>>>>>an
> >>>>>>>>N-dimensional space, with each attribute being orthogonal to all of
> >>>>>>>>the
> >>>>>>>>others.
>
> >>>>>>>What does that mean ? How is a single attribute orthogonal to N-1
> >>>>>>>attributes part of the same relation?. What do you exactly designate
> >>>>>>>as *orthogonality*? Why would a tuple necessarily be a point and not
> >>>>>>>a line ofr plane in geometrical N-space?
> >>>>>>>What about degree 0/1 relations? How does a degree 0 relation
> >>>>>>>represent a point in space?
>
> >>>>>>The answers to all your questions can be found in any linear algebra
> >>>>>>textbook.
>
> >>>>>Really? Would you care providing a source?
>
> >>>>"If a relation has /n/ columns, then /each row in that relation
> >>>>represents a
> >>>>point in n-dimensional space/--and the relation as a whole represents a
> >>>>set
> >>>>of such points. In other words, a relation of /n/ columns is
> >>>>/n/-dimensional, not two-dimensional. *Let's all vow never to say "flat
> >>>>relations" ever again.*"
>
> >>>>"Date on Database", ISBN 1-59059-746-X, page 371
>
> >>>>As for a linear algebra textbook, there are thousands of them. Pick one
> >>>>and
> >>>>make the necessary extrapolation to relational database theory.
>
> >>>That's what I suspected (Date's algebra).
>
> >>>Thank you anyway for providing some sources (I do appreciate the
> >>>effort). The purpose of my questions was to raise the issue that one
> >>>may *arbitrarily choose several math constructs to describe the *non
> >>>flat* nature of relations. Choosing a POINT in N dimensionnal space
> >>>(correlated to N attributes) to describe relation tuple set is just
> >>>one *possible* math construct choice and such choice has implications
> >>>I never felt confortable with. If I apply Date's definition replacing
> >>>the *n* by some values:
>
> >>>>A degree 1 relation tuple set is necessarily a LINE (1 - dimensional
> >>>>space)
> >>>>A degree 2 relation tuple set is necessarily represented as a PLANE (2 -
> >>>>dimensional space)
> >>>>How is a degree 0 relation tuple set represented considering that it has
> >>>>0-dimensions according to Date?
>
> >> Good point! (Please excuse the pun; an opportunity like this arises only
> >> once in a lifetime. :-) )
>
> >> What /does/ an empty tuple represent? Is it some sort of NULL, as opposed
> >> to a null /set/?
>
> >> And since the tuples in a relation must all be distinct, that would seem
> >> to imply that a non-empty relation having no attributes could contain
> >> only a single "null tuple". Could this "null relation" serve as an
> >> identity element for joins?
>
> >> Let me get back to you ... :-)
>
> > Not null. True. A non-empty 0-ary relation is the relational equivalent of
> > True. An empty 0-ary relation is the relational equivalent of False.
>
> Allow me to correct a poor word choice in my previous post: When I wrote
> NULL, I was thinking along the lines of a null reference, as in C or C#, NOT
> a SQL NULL.
>
> I think I see where you're going with True and False as defined in your
> post, though ...
So, let me summarize this, based on D&D algebra, degree N relation tuples are points located in N-coordinate system. Therefore, degree-1 relation tuples set make a LINE , degree-2 relation tuple set makes a PLANE and so forth...As degree-0 is described in D&D as equivalent TRUE (if there are tuples) and FALSE (if empty), do you see how the POINT analogy becomes ambiguous? In my perspective such analogy had the sole purpose of explaining that in no case relations are to be mistaken with tables/ I have not read yet any formal demonstration qualifying degree-0 relation according to such hypothesis. Received on Thu Jul 26 2007 - 23:24:57 CEST

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