Re: A different definition of MINUS, Part 3

On Dec 21, 9:14 pm, paul c <toledobythe...@xxxxxxxx> wrote:
vadim...@xxxxxxxxx wrote:

...

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. ...

In other words, D&D has absorption when projection is applied and its
union allows deMorgan.  RL has absorption without projection but
projection is defined in terms of a second kind of union.  They both
have distributivity and associativity and defined identity values

Not quite. Both distributivities of ^ over v, and v over ^ are
conditional. Distributivities of + over ^, and ^ over + are universal.

although D&D needs only two identities.  

They don't need additional constants because they don't define
negation. In RL negation is defined with two axioms:

x' ^ x = x ^ R00.
x' v x = x v R11.

Double negation, and De Morgan

x' + y' = (x ^ y)'

are theorems in RL.

As for RL and deMorgan, I
thought Marshall S said at least a year ago that RL supported deMorgan.
  I'm not sure, does it?

x' v y' = (x ^ y)'

is not valid in RL, example

y = {<q=a,>,}
x = {<p=1,>,}

(I changed attribute names in QBQL to not collide with relation names)

.

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