Re: A different definition of MINUS, Part 3

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. Received on Mon Dec 22 2008 - 07:22:42 CET