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Inside Microsoft SQL Server 2008: T-SQL Querying
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If {A1, A2, . . ., Ak} is a partition of S, the answer to In which Ai is the element x is well-de ned. We ve already seen one example of a partition: Given a set S with universe U, the sets S and SC partition U.
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Generalizations of Set Theory
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An understanding of basic set theory is a great help, but it s important to recognize its limitations in describing the world, and in the case of this book, T-SQL querying. I ve already addressed some of the ways in which mathematics fails to represent the world precisely, but one generalization of set theory is particularly relevant to databases.
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Multiset Theory
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It s a mathematical fact that the sets {2, 8, 4, 4}, { 4, 8, 4, 2}, and {2, 4, 2, 8, 4, 8, 2, 4, 4} are equal, but you would probably agree that the last set listed contains three twos. Of course, a set S of numbers can t contain three twos. It can either contain a two or not contain a two. If it contains a two, 1S(2) = 1. If it doesn t, 1S(2) = 0, and nothing else is possible. It s possible to accommodate the idea of multiple membership in set theory, except that it would no longer be set theory, it would be multiset theory, sometimes known as the theory of bags. The simplest way to begin developing a theory of multisets is by generalizing the characteristic function.
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The Multiplicity Function of a Bag
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If B is a bag (or multiset) with universe U, the multiplicity function of B, denoted MB, is the function on U that tells how many copies of an element B contains.
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Many de nitions from set theory extend almost unchanged to bag theory if the characteristic function is replaced by the multiplicity function. For example, the multiplicity function of an intersection can be taken to be the minimum of the multiplicity functions. Other notions are far more problematic. It s not clear how to de ne a multiset s complement, for example. Should universal sets contain an unlimited number of each of their elements, and should the complement of any nite multiset be in nite Because set cardinalities have more than one size of in nity, which size should be used for multisets The problems with multiset theory often lead database theoreticians to outlaw duplicate rows within a table for example, by requiring primary key constraints. It s harder to prevent result sets from containing duplicates, however. This would require changing the meaning of SELECT to what is now written as SELECT DISTINCT, and this would create other complications, particularly with aggregates. T-SQL, like most SQL dialects, supports multisets
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Set Theory and Predicate Logic
in most places but not everywhere. T-SQL, for example, doesn t support EXCEPT ALL and INTERSECT ALL, only EXCEPT DISTINCT and INTERSECT DISTINCT.
Predicate Logic
Predicate logic is a mathematical framework for representing and manipulating expressions that are true or false: facts and falsehoods.
Logic-Like Features of Programming Languages
T-SQL, like many programming languages, incorporates true-false expressions and logical operators in several places, not all of which are, strictly speaking, related to predicate logic.
Note A true-false expression is called a Boolean expression (after the logician George Boole).
Boolean logic begins with the study of Boolean expressions.
The Keyword IF in Control-of-Flow Statements
Although the focus of this book is on T-SQL s query language, and SQL s central (or at least most interesting) paradigm is set based, regular programming based on decision and repetition is also implemented. For example, many of this book s code samples begin with a conditional statement to delete an object if it already exists. You encountered this statement in 1, Logical Query Processing :
IF OBJECT_ID('dbo.Orders') IS NOT NULL DROP TABLE dbo.Orders;
This is a valid T-SQL statement, and it conforms to the syntax SQL Server Books Online gives for an IF. . .ELSE statement:
IF Boolean_expression { sql_statement | statement_block } [ELSE { sql_statement | statement_block } ]
The Boolean expression is OBJECT_ID( dbo.Orders ) IS NOT NULL, and the sql_statement is DROP TABLE dbo.Orders. The way in which a program implements decision making or repetition is often referred to as the program s logic. Formal logic, however, isn t about what happens when a program runs, nor is it about the way in which programs implement algorithms to solve problems. In particular, the expression if <this> then <that> in formal logic bears nothing more than a super cial resemblance to the statement IF <this> THEN <that> in a programming language. The former is a sentence that in its entirety is either true or false; the latter is an instruction to produce behavior that depends on whether <this> (not the entire statement) is true or false.
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