Re: a union is always a join!

Brian Selzer wrote:

>>> The mathematics of relational calculus and relational algebra are fully
>>> capable, AFAIK, of describing the difference between two states of a
>>> database.

[...]

>The algebra is capable of that if and only if each and every tuple has a key
>that permanently identifies the thing in the universe of discourse that the
>tuple maps to.

No, the algebra can describe the difference between database states without any assumption on how these states are interpreted. And as Walter Mitty already wrote, it is not in fact necessary that databases are inpeepreted in such a way that tuples are about "things" in the universe of discourse.

I think you are overstating your point, which was, if I understand you correctly, that while relational algebra and calculus may be used to express the contents of a database change, they do not express the fact that the contents change; and this does need to be expressed in some way when reasoning about database updates.

>But since that isn't a requirement of the RM, the RM must be
>in trouble! If different keys at different states map to the same thing in
>the universe of discourse, or if the same key at different states maps to
>different things in the universe of discourse, then how is it possible to
>describe exactly what is different between two states of a database.

It is really simple. However, you are right in that mutability of keys and other aspects of the relationship between database contents and its interpretation will fall outside the scope of that description.

-- 
Reinier