# Re: A different definition of MINUS, Part 3

Date: Mon, 22 Dec 2008 09:23:14 -0800 (PST)

Message-ID: <eea8555e-15bc-407c-8ff3-8199fac61c33_at_q26g2000prq.googlegroups.com>

On Dec 21, 10:36 pm, Bob Badour <bbad..._at_pei.sympatico.ca> wrote:

> vadim..._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 + ?
*

This was a typo.

http://arxiv.org/ftp/arxiv/papers/0807/0807.3795.pdf
page 5
Received on Mon Dec 22 2008 - 18:23:14 CET