SQL and Relational Theory, 2nd Edition

PROJECTION

Definition: Let r be a relation and let A, B, ..., C be attributes of r. Then the projection of r on (or over) those attributes, r{A,B,...,C}, is a relation with (a) heading {A,B,...,C} and (b) body the set of all tuples x such that there exists some tuple t in r with A value equal to the A value in x, B value equal to the B value in x, ..., and C value equal to the C value in x.

For example:

To repeat, the result is a relation; thus, “duplicates are eliminated,” to use the common phrase, and that DISTINCT in the SQL formulation is really needed, therefore.^[76] The result heading has attributes (or columns) COLOR and CITY—in that left to right order, in SQL.

Let r be a relation. Then:

• The projection r{H}, where {H} is all of the attributes—in other words, the heading—of r, just returns r. Such a projection is known as an identity projection.

• The projection r{}—in other words, the projection of r on no attributes at all—returns TABLE_DEE if r is nonempty, TABLE_DUM otherwise. Such a projection is sometimes called a nullary projection; however, the term nullary is best avoided because of the potential confusion with SQL-style nulls. (Just to remind you, TABLE_DEE is the unique relation with no attributes and just one tuple—the 0-tuple, of course—and TABLE_DUM is the unique relation with no attributes and no tuples at all. The fact that projecting r on no attributes always yields ...