SQL and Relational Theory, 2nd Edition

RELATIONAL COMPARISONS

Like tuple types, relation types are no exception to the rule that the “=” comparison operator must be defined for every type; that is, given two relations r1 and r2 of the same relation type T, we must at least be able to test whether they’re equal. Other comparisons might be useful, too; for example, we might want to test whether r1 includes r2 (meaning every tuple in r2 is also in r1), or whether r1 properly includes r2 (meaning every tuple in r2 is also in r1 but r1 contains at least one tuple that isn’t in r2). Here’s an example, expressed in Tutorial D as usual, of an equality comparison on relations:

     S { CITY } = P { CITY }

The left comparand here is the projection of suppliers on {CITY},^[45] the right comparand is the projection of parts on {CITY}, and the comparison returns TRUE if these two projections are equal, FALSE otherwise. In other words, the comparison (which is a boolean expression) means: “The set of supplier cities is equal to the set of part cities” (and it evaluates to either TRUE or FALSE, of course).

Here’s another example

     S { SNO } ⊃ SP { SNO }

The symbol “⊃” here means “properly includes” (or, equivalently, “is a proper superset of”). The meaning of this expression (considerably paraphrased) is: “Some suppliers supply no parts at all” (which again necessarily evaluates to either TRUE or FALSE).

Other useful relational comparison operators include “⊇” (“includes”), “⊆” (“is included in”), and “⊂” (“is properly included in”). Note: Of ...