# Re: A different definition of MINUS, Part 3

Date: Sun, 21 Dec 2008 22:34:10 -0800 (PST)

Message-ID: <d608da87-7f55-4a2c-bd01-69980128338a_at_n33g2000pri.googlegroups.com>

On Dec 21, 10:22 pm, 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
*

--------------^^^^^^^^^

Typo: <OR>

*> can be represented
*

> *algebraically* as

*>
**> x <AND> y ≝ (x ^ (y v R11)) v (y ^ (x v R11)).
*

--------------^^^^^^^^^

Ditto

*>*

> This operator can be proven enjoying some nice properties, including

*> DeMorgan.*

Received on Mon Dec 22 2008 - 07:34:10 CET