Re: A different definition of MINUS, Part 3
From: Bob Badour <bbadour_at_pei.sympatico.ca>
Date: Sun, 21 Dec 2008 23:18:42 -0400
Message-ID: <494f0715$0$5493$9a566e8b_at_news.aliant.net>
>
>
> OK, in classic relational algebra union can only be applied to the
> relations with the same header (that is set of attributes). Therefore,
> when generalizing union to become applicable to any pair of relations
> one must decide first, what the header the resulting relation should
> have. D&D assumed it has to also be a union, but I suggest that it can
> be anything: intersection, difference, or even symmetric difference.
> However, the last two choices are no good: symmetric difference would
> make the generalized version of the union incompatible with classic RA
> union, while difference operation is not symmetric, thus rendering
> generalized union nonsymmetric as well. Therefore, the only
> alternative to D&D version of the union is "inner union": it
> intersects over headers, and unions over tuples. Compare it to join
> that intersects on tuple level, and unions headers.
>
> Next one may compare D&D <AND>&<OR> based system, with RL join&inner
> union based one in terms of consistency. Both have arguments in their
> favor. D&D system honors distributivity, and De Morgan laws. RL honors
> absorption, so that the subset relation can be generalized to be
> applicable to any pair of relations. Also RL can express projection as
> an (inner) union of a relation with an empty relation. First, tuples
> in both relations (there are none in the second!) are collapsed to the
> common set of attributes. These are essentially projections. Then we
> make a union of projections, but keep in mind that the second
> projection is empty!
Date: Sun, 21 Dec 2008 23:18:42 -0400
Message-ID: <494f0715$0$5493$9a566e8b_at_news.aliant.net>
vadimtro_at_gmail.com wrote:
> On Dec 21, 4:13 pm, Bob Badour <bbad..._at_pei.sympatico.ca> wrote:
>
>>Projection, to me, doesn't seem like any sort of union.
>
>
> OK, in classic relational algebra union can only be applied to the
> relations with the same header (that is set of attributes). Therefore,
> when generalizing union to become applicable to any pair of relations
> one must decide first, what the header the resulting relation should
> have. D&D assumed it has to also be a union, but I suggest that it can
> be anything: intersection, difference, or even symmetric difference.
> However, the last two choices are no good: symmetric difference would
> make the generalized version of the union incompatible with classic RA
> union, while difference operation is not symmetric, thus rendering
> generalized union nonsymmetric as well. Therefore, the only
> alternative to D&D version of the union is "inner union": it
> intersects over headers, and unions over tuples. Compare it to join
> that intersects on tuple level, and unions headers.
>
> Next one may compare D&D <AND>&<OR> based system, with RL join&inner
> union based one in terms of consistency. Both have arguments in their
> favor. D&D system honors distributivity, and De Morgan laws. RL honors
> absorption, so that the subset relation can be generalized to be
> applicable to any pair of relations. Also RL can express projection as
> an (inner) union of a relation with an empty relation. First, tuples
> in both relations (there are none in the second!) are collapsed to the
> common set of attributes. These are essentially projections. Then we
> make a union of projections, but keep in mind that the second
> projection is empty!
Okay, you seem to be saying that DeMorgan holds for D&D but not for RL. Didn't you use DeMorgan in the proof that amazed you? Was that D&D or RL? Received on Mon Dec 22 2008 - 04:18:42 CET