SQL and Relational Theory, 2nd Edition

WHAT DO RELATIONAL EXPRESSIONS MEAN?

Recall now from Chapter 5 that every relvar has a certain relvar predicate, which is, loosely, what the relvar means. For example, the predicate for the suppliers relvar S is:

Supplier SNO is under contract, is named SNAME, has status STATUS, and is located in city CITY.

What I didn’t mention in Chapter 5, however, is that the foregoing notion extends in a natural way to apply to arbitrary relational expressions. For example, consider the projection of suppliers on all attributes but CITY

     S { SNO , SNAME , STATUS }

This expression denotes a relation containing all tuples of the form

     TUPLE { SNO s , SNAME n , STATUS t }

such that a tuple of the form

     TUPLE { SNO s , SNAME n , STATUS t , CITY c }

currently appears in relvar S for some CITY value c. In other words, the result represents the current extension of a predicate that looks like this (see Chapter 5 if you need to refresh your memory regarding the notion of a predicate’s extension):

There exists some city CITY such that supplier SNO is under contract, is named SNAME, has status STATUS, and is located in city CITY.

This predicate thus represents the meaning of the relational expression S{SNO,SNAME,STATUS}. Observe that it has just three parameters and the corresponding relation has just three attributes—CITY isn’t a parameter to that predicate but what logicians call a “bound variable” instead, owing to the fact that it’s “quantified” by the phrase There exists some city (see Chapter 10 for further ...