Re: A different definition of MINUS, Part 3
From: Bob Badour <bbadour_at_pei.sympatico.ca>
Date: Mon, 22 Dec 2008 02:36:45 -0400
Message-ID: <494f357f$0$5464$9a566e8b_at_news.aliant.net>
>>vadim..._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?
Date: Mon, 22 Dec 2008 02:36:45 -0400
Message-ID: <494f357f$0$5464$9a566e8b_at_news.aliant.net>
vadimtro_at_gmail.com wrote:
> On Dec 21, 7:18 pm, Bob Badour <bbad..._at_pei.sympatico.ca> wrote: >
>>vadim..._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?
> > > In RL the <AND> operator is not considered fundamental operator, and > can be represented > *algebraically* as > > x <AND> y ≝ (x ^ (y v R11)) v (y ^ (x v R11)). > > This operator can be proven enjoying some nice properties, including > DeMorgan.
When you used DeMorgan, I didn't think you were using <AND>. Weren't you using ^ and + ? Received on Mon Dec 22 2008 - 07:36:45 CET