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

From: Bruce C. Baker <bcb_at_undisclosedlocation.net>
Date: Thu, 26 Jul 2007 15:53:25 -0500
Message-ID: <bR7qi.9286\$lZ7.204_at_newsfe20.lga>

> Bruce C. Baker wrote:
>
>> "Cimode" <cimode_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 ... Received on Thu Jul 26 2007 - 22:53:25 CEST

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