Re: Relation subset operators

From: <cimode_at_hotmail.com>
Date: Sat, 6 Jun 2009 11:43:12 -0700 (PDT)

On 6 juin, 20:02, paul c <toledobythe..._at_oohay.ac> wrote:
> cim..._at_hotmail.com wrote:
>
> ...
>
> > In a way that gave to think that relational model is *not* ready yet
> > to apply absurd mathematical reasonning yet.  Be it  finally sound or
> > not, absurd logical reasonning is too risky for a young science to be
> > used as a foundation .  I will stick to basic clarifications on a step
> > y step basis and let the big theories to smarter people.
> > ...
>
> No argument with that, but if we're talking about a shorthand for
> division, which is universal quantification, which is defined in fopc,
> then the shorthand should produce an equivalent answer to fopc, eg., as
> Date says:
>
> (quote)
> Have you ever wondered why the operation is called "division?" The
> following identity shows why:
>
> ( R TIMES S ) DIVIDEBY S = R
>
> Division is a kind of inverse of Cartesian product, and it isn't quite
> the counterpart to the universal quantifier that it was meant to be. It
> suffers from problems having to do with empty sets and related matters.6
> In fact, Codd offers an example that illustrates the problem: Given a
> relation SP { S#, P#, ... } showing which suppliers supply which parts,
> he claims that the expression SP { S#, P# } DIVIDEBY SP { P# } will give
> numbers for suppliers who provide all parts. However, if there aren't
> any parts, this expression gives the wrong answer. (It gives no supplier
> numbers, yet it should give them all).
Thank you for this reminder.

> Relational comparisons provide a better basis for dealing with the kinds
> of problems that division was intended to solve -- but the relational
> model as originally defined by Codd didn't include such comparisons at
> all.7
I would go further than that into saying that previous work has only clarified side effects of relation operations. And a lot of it missed the mark into expressing properties of operations that can not be expressed without proper quantifiers. For instance, does the empty set has the same role place in relational theory, than the zero would have in traditional algebra. Up till, such questions have not been answered and these claims have neither been properly demonstrated nor they have been properly evaluated. However nothing prevented demonstrators of using such quantifier in relational operations. There is something in that puzzles me. The creators of the zero did prove and demonstrate the usefulness of such value into simplifying algebra before they could actually use it. Nothing similar can be said of all particular relation that have been created in packs and used (Empty sets, DEE, DUM etc...)...In a word, a lot of demonstrations were made using tools but nobody questionned the relevance of such tools before using them...That is a deep sign of immaturity. Received on Sat Jun 06 2009 - 20:43:12 CEST

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