Relational Theory for Computer Professionals

Appendix B. TABLE_DUM and TABLE_DEE

Tweedledum and Tweedledee Agreed to have a battle; For Tweedledum said Tweedledee Had spoiled his nice new rattle. Just then flew by a monstrous crow As big as a tar-barrel; Which frightened both the heroes so, They quite forgot their quarrel.

—old English nursery rhyme, quoted by Lewis Carroll in Through the Looking-Glass and What Alice Found There (1871)

As noted in Chapter 3, the empty set—the set that contains no elements—is a subset of every set. It follows in particular that the empty heading is a valid heading, and hence that a tuple with an empty set of components is a valid tuple. Such a tuple is of type TUPLE { }; indeed, it’s sometimes referred to explicitly as a 0-tuple, in order to emphasize the fact that it has no components and is of degree zero. It’s also sometimes called an empty tuple. In Tutorial D, it’s denoted thus:

TUPLE { }

Note: The syntactic construct TUPLE { } thus does double duty in Tutorial D. However, the intended interpretation is always clear (and unambiguous) from context.

It follows further from the above that a relation too can have an empty heading. Such a relation is of type RELATION { }, and its degree is zero. In Tutorial D, such a relation can be denoted thus:

RELATION { } body

Here the braces denote the empty heading, and body is either { } or {TUPLE{ }}—see the explanation immediately following, also Appendix A.

Let r be a relation of degree zero, then. How many such relations are there? The answer is: Just two. ...