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CHAPTER 8 WHAT FIRST NORMAL FORM REALLY MEANS
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and fractional parts (say). But even if we were to accept this argument, it seems to me we would then also have to accept the argument that the very same value might be atomic today and nonatomic tomorrow! Such would be the case for a fixed-point number, for example, if there were originally no operators to extract its integer and fractional parts but such operators were subsequently introduced. And it further seems to me that if the notion of atomicity is timedependent in this way, then (as already claimed) it really doesn t have any absolute meaning.
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Let s return for a moment to table T2 in Figure 8-2. In that figure, I showed P#_SET values as general sets. But it s much more useful in practice if they re, more specifically, relations (see Figure 8-3, where I ve renamed the column as P#_REL and the table accordingly as T3). Why is it more useful Because relations, not general sets, are what the relational model is all about.5 As a consequence, the full power of the relational algebra immediately becomes available for the relations in question they can be restricted, projected, joined, and so on. By contrast, if we use general sets instead of relations, then we need to introduce new operators (set union, set intersection, and so on) for dealing with those sets ... Much better to get as much mileage as we can out of the operators we already have!
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5. In case you re wondering, the difference is that general sets can contain anything, whereas relations contain tuples specifically. Note, however, that a relation certainly resembles a general set in that it too can be regarded as a single value.
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CHAPTER 8 WHAT FIRST NORMAL FORM REALLY MEANS
RVA is the usual abbreviation, which is why I m now talking in terms of relations and attributes instead of tables and columns. For present purposes, you can regard the terms as interchangeable (but see Appendix A). If A is an RVA and D is the domain underlying A, then the values in D are relations too, of course. In general, the relations that are values of some given RVA can have attributes that are RVAs in turn, and so we can have relations that contain other relations nested inside themselves, and those other relations can contain still further relations nested inside them, and so on, recursively, to any number of levels. All without violating 1NF! Just as an aside, I remind you that, as noted earlier, the 6th edition of my book An Introduction to Database Systems said (paraphrasing) that 1NF meant no repeating groups. But it also said that 1NF allowed RVAs and you might feel these two statements contradict each other somewhat. In fact, however, I was attempting to draw a distinction between repeating groups, which weren t very precisely defined, and RVAs, which were. With hindsight, I think that attempt was probably a mistake on my part. (Though possibly not, if repeating groups still aren t precisely defined. See later in this chapter for further discussion of this latter point.) Of course, we need a way to map between tables without RVAs (like table T1) and tables with them (like table T3), and that s the purpose of the GROUP and UNGROUP operators of the relational algebra. I don t want to go into too much detail on those operators here; let me just say that, given tables T1 and T3 as shown in Figs. 1 and 3, respectively, the expression T1 GROUP ( { P# } AS P#_REL ) will produce table T3, and the expression T3 UNGROUP ( P#_REL ) will produce table T1. Note: All coding examples in this chapter are expressed in a language called Tutorial D, which is described in detail in a book by Hugh Darwen and myself, Databases, Types, and the Relational Model: The Third Manifesto, 3rd edition (Addison-Wesley, 2006) referred to hereinafter as just The Third Manifesto. In the interests of familiarity I would have preferred to use SQL (though it s not much to my taste); however, SQL doesn t support all of the constructs I need to discuss, so I m more or less forced into using a language of my own. This state of affairs is a little unfortunate, but I believe the examples are all more or less selfexplanatory anyway. There are a few more points I d like to make in connection with GROUP and UNGROUP. First of all, Codd himself in fact proposed a grouping operator he called it factoring in an appendix to his paper Relational Completeness of Data Base Sublanguages, which appeared in the same book as the further normalization paper mentioned near the beginning of the previous section. I don t think he regarded that operator as part of the relational algebra, however, precisely because it produced a result with an RVA. His actual words were: For presentation purposes, it may be desirable to convert a normalized relation to unnormalized form. The operation of factoring accomplishes this. In other words, Codd saw factoring as converting a relation into something that wasn t a relation but might perhaps be called a report (note the words For presentation purposes ).
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