# 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:

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