SQL and Relational Theory, 2nd Edition

TABLE_DUM AND TABLE_DEE

Recall from the discussion of tuples earlier in this chapter that the empty set is a subset of every set, and hence that there’s such a thing as the empty tuple (also called the 0-tuple), and of course that tuple has an empty heading. For exactly the same reason, a relation too might have an empty heading—a heading is a set of attributes, and there’s no reason why that set shouldn’t be empty. Such a relation is of type RELATION {}, and its degree is zero.

Let r be a relation of degree zero, then. How many such relations are there? The answer is: Just two. First, r might be empty (meaning it contains no tuples)—remember there’s always exactly one empty relation of any given type. Second, if r isn’t empty, then the tuples it contains must all be 0-tuples. But there’s only one 0-tuple!—equivalently, all 0-tuples are duplicates of one another—and so r can’t possibly contain more than one of them. So there are indeed just two relations with no attributes: one with just one tuple, and one with no tuples at all. For fairly obvious reasons, I’m not going to try drawing pictures of these relations (in fact, this is the one place where the idea of thinking of relations as tables breaks down completely).

Now, you might well be thinking: So what? Why on earth would I ever want a relation that has no attributes at all? Even if they’re mathematically respectable (which they are), surely they’re of no practical significance? In fact, however, it turns out they’re of very ...