Applied Mathematics for Database Professionals

Applied Mathematics for Database Professionals

Lex de Haan, Toon Koppelaars

Language: English

Pages: 405


Format: PDF / Kindle (mobi) / ePub

Relational databases hold data, right? They indeed do, but to think of a database as nothing more than a container for data is to miss out on the profound power that underlies relational technology. A far more powerful way of thinking lies in relational technologies foundation in the mathematical disciplines of logic and set theory.

Databases contain truths or propositions describing some area of interest such as a business. Those truths are organized into sets. Operations from logic and set theory can be applied to existing sets of truths to derive new sets of truths. Applied Mathematics for Database Professionals introduces you to this way of thinking, to the logic and set theory that underlies relational database technology. All this may sound abstract now, but there are profound benefits from the deeper understanding you'll gain from this book. You'll learn to
* Become a better database designer. You'll make fewer mistakes, and your designs will be more flexible in response to changing data needs.
* Use the expressive power of mathematics to precisely specify designs and business rules.
* Communicate effectively about design using the universal language of mathematics.
* Develop and write complex SQL statements with confidence.
* Avoid pitfalls and problems from common relational bugaboos such as null values and duplicate rows.

The math that you learn in this book will put you above the level of understanding of most database professionals today. You'll better understand the technology and be able to apply it more effectively. You'll avoid data anomalies like redundancy and inconsistency. Understanding whats in this book will take your mastery of relational technology to heights you may not have thought possible.

This book is reviewed and endorsed by C. J. Date and features a foreword by the same.

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one logical expression into another (equivalent) logical expression. This chapter is an introductory chapter on logic. Chapter 3 will continue where this one stops—the two chapters make up one single topic (logic). The split is necessary because some concepts concerning logic require the introduction of a few set-theory notions first. Chapter 2 will serve that purpose. The introduction of the crucial concept of rewrite rules at the end of this chapter opens up the first possibility to do some

Q P|Q T T F T F T F T T F F T Note that this is purely a theoretical exercise, exploring the extreme edges of our game; although you can indeed rewrite all possible logical expressions using this single NAND connective, your expressions will become longer and much more difficult to read. Table 1-10 shows how you can express NOT using NAND by using the same propositional variable (P) for the left and right operands of the NAND connective. 7451CH01.qxd 5/4/07 1:38 PM Page 17

7451CH06.qxd 5/14/07 10:37 AM Page 121 CHAPTER 6 ■ TUPLE, TABLE, AND DATABASE PREDICATES In the same way as with tuple predicates, we say that P5 is a table predicate over {partno}, P6 is a table predicate over {instock,price}, and P7 is a table predicate over {instock,partno, name}. Now take a look at table PAR1 in Figure 6-1. Figure 6-1. Table PAR1 PAR1 is a table over {partno,name,instock,price}. This is a superset of all sets that P5, P6, and P7 are table predicates over; we can

this chapter: to define a database universe. In this phase you formally specify the database (multi-table) constraints. Because the example database universe presented in this chapter has ten table structures, we’ll introduce you to ten characterizations, ten tuple universes, and ten table universes. This, together with the explanatory material provided, makes this chapter a rather big one. However, the number of examples should provide you with a solid head start to applying the formal theory,

already been introduced to this; in this phase all characterizations (one per table structure) are defined. 141 7451CH07.qxd 142 5/15/07 9:43 AM Page 142 CHAPTER 7 ■ SPECIFYING DATABASE DESIGNS You then use the characterizations as building blocks to build (define) for each table structure the set of admissible tuples. This involves applying the generalized product operator (see Definition 4-7) and the introduction of tuple predicates. The set of admissible tuples is called a tuple

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