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 great practical significance indeed: so much so, that we have pet names for them—we call them TABLE_DUM and TABLE_DEE, or DUM and DEE for short (DUM is the empty one, DEE is the one with one tuple). And what makes them so significant is their meanings, which are FALSE (or no) for DUM and TRUE (or yes) for DEE. They have the most fundamental meanings of all. Note: I’ll be discussing the whole notion of relations and their meaning in much more detail in Chapter 5 and Chapter 6.

By the way, a good way to remember which is which is this: DEE and yes both have an “E”; DUM and no don’t.

Now, I haven’t covered enough in this book yet to show concrete examples of DUM and DEE in action, as it were, but we’ll see plenty of examples of their use in the pages ahead. Here I’ll just mention one point that should make at least intuitive sense at this early juncture: These two relations (especially TABLE_DEE) play a role in the relational algebra that’s analogous to the role played by zero in conventional arithmetic. And we all know how important zero is; in fact, it’s hard to imagine an arithmetic without zero (the ancient Romans tried, but it didn’t get them very far). Well, it should be equally hard to imagine a relational algebra without TABLE_DEE. Which brings us to SQL ... SQL, since it has no counterpart to the 0-tuple, clearly (but unfortunately) has no counterpart to TABLE_DUM or TABLE_DEE either.^[46]