Re: A different definition of MINUS, Part 3

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

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> can be represented
> *algebraically* as
>
> x <AND> y ≝ (x ^ (y v R11)) v (y ^ (x v R11)).

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