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CHAPTER 12 DATA REDUNDANCY AND DATABASE DESIGN
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Tuples vs. Propositions
Let me begin by reminding you that every tuple appearing in relvar R at a given time represents a certain proposition, where the proposition in question is an instantiation of the relvar predicate for R that (by convention) is understood to be true at the time in question. In the case of relvar HP from Figures 12-5 and 12-6, for example, the relvar predicate is: Part P# is used in the enterprise, is named PNAME, has color COLOR and weight WEIGHT (which is greater than or equal to 17 pounds), and is stored in city CITY. And the tuple for part P6 (for example) represents the following instantiation of that predicate: Part P6 is used in the enterprise, is named Cog, has color Red and weight 19.0 (which is greater than or equal to 17 pounds), and is stored in city London.
CHAPTER 12 DATA REDUNDANCY AND DATABASE DESIGN
I also suggested earlier that the database involves some redundancy if and only if it says the same thing twice. Now I can make this statement a little more precise: The database involves some redundancy if and only if it contains two representations of the very same proposition. Now, given that tuples represent propositions, it s tempting to translate this latter statement into the following one: The database involves some redundancy if and only if it contains two appearances of the very same tuple. Unfortunately, this definition is, at best, considerably oversimplified. Let s examine it more carefully. First of all, it s at least true that we don t want the same tuple to appear more than once within the same relation (and here I do mean relation, not relvar), because such a state of affairs would certainly constitute saying the same thing twice. (As I once heard Codd remark: If something is true, saying it twice doesn t make it any more true.) Of course, the relational model itself takes care of this requirement by definition, relations never contain duplicate tuples.7 As an aside, I observe that we now have a precise characterization of the notion of duplicate tuples (people use this phrase all the time, and yet I very much doubt whether many of them would be able to define it precisely if pressed). Strictly speaking, of course, two tuples are duplicates if and only if they re the very same tuple just as two integers are duplicates if and only if they re the very same integer and the phrase duplicate tuples thus doesn t really make much sense from a logical point of view. What people really mean when they use that phrase is duplicate appearances of the same tuple recall from the section Some Prerequisites that value vs. appearance is one of the great logical differences (tuples are certainly values, of course). To get back to the main discussion: Second, it s easy to see that the very same tuple can represent any number of distinct propositions. As a trivial example, let SCITY and PCITY be the projections of suppliers on CITY and parts on CITY, respectively. Given the sample data values of Figure 12-1, then, the tuple <London> appears in both SCITY and PCITY, but it represents two different propositions in SCITY, it represents the proposition There s at least one supplier in London, in PCITY it represents the proposition There s at least one part in London (simplifying slightly for the sake of the example). What s more and here I have to get a little more formal on you for a moment the same proposition can be represented by any number of distinct tuples, too. That s because, formally, the pertinent attribute names are part of the tuple. Thus, for example, we might have our usual relvar SP with its attributes S#, P#, and QTY, and predicate: Supplier S# supplies QTY of part P#. We might additionally have a relvar PS with attributes SNO, PNO, and AMT, with predicate: Supplier SNO supplies AMT of part PNO. And then (to use Tutorial D syntax) the following tuples might appear in these two relvars, respectively:
7. In other words, it might be argued that a desire to avoid redundancy is one of the motivations (perhaps a minor one) for choosing sets which don t contain duplicate elements, by definition instead of bags, which do, as the right mathematical abstraction on which to found a solid database theory. SQL apologists please note!
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