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Relational Algebra and Relational Calculus
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To manipulate relations (relational variables), you need some operators. Relations and operators on relations form what is called relational algebra. The collection of relational operators is open ended, but some operators are considered basic. Although the basic operators are somewhat intuitive, I ll introduce them for the sake of completeness.
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Basic Operators
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As for simple types, we need at least two operators on relation types: one to store a set of tuples in a relational variable and one to retrieve a variable s value. These correspond to the familiar notions of assignment and evaluation. The relational selector operator (corresponding to evaluation) returns a table from a relational variable, and the relational assignment operator assigns a table value to a relational variable. A set of basic Boolean operators on relations and tuples is obviously needed as well:
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= (equals) (not equals) (subset of) (superset of) (element of) = (is empty)
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The Relational Model
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The rst four operators listed here accept two relations as parameters. The fth one checks whether a tuple is a member of a relation in other words, it accepts a tuple as the left parameter and a relation as the right parameter. Finally, the last operator in the list accepts a single relation as a parameter and checks whether it is empty. If you wish, you can de ne additional operators for convenience, such as proper subset of (to mean subset of and not equal to) and proper superset of (superset of and not equal to). I want to mention one other speci c operator that helps greatly with the tabular presentation of a relation the Order By <attribute_1,attribute_2,. . .,attribute_n> operator. The Order By operator does not return an unordered result; thus, it does not return a set or relation, which are unordered. You can think of the return value of the Order By operator as a sorted table. Sorting is not prede ned for relations and tuples, however; therefore, supporting the Order By operator for a particular relation requires that at least one attribute of the relation support ordering and the following operators:
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> (greater than) (greater than or equal to) (less than or equal to) < (less than)
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The table returned by the Order By operator is sorted according to values of one or more attributes, all of which must be of data types that support the listed type operators.
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Relational Algebra
Relational algebra is a collection of operators that accept relations as input parameters and return relations. The fact that the result of any relational operation is a relation is referred as the relational closure property of the relational algebra. Codd originally de ned eight relational operators four of them are based on traditional set operators, and four of them are special relational operators. These eight are Restrict, Project, Product, Union, Intersect, Minus, Join, and Divide. Relational algebra is not closed; you can de ne additional operators as long as they respect the relational closure property. I ll introduce a handful of useful operators in addition to Codd s original eight. Of course, because the collection of relational operators is open ended, my list is not complete. I deliberately selected the operators that I nd most useful and that are used in the Transact-SQL language later in this book.
Codd s Eight Original Operators
The Restrict operator lters tuples of a relation. The result of this operator is a relation with fewer tuples than (or the same number as) the original relation. The heading type of the relation returned is the same as the heading type of the original relation. The restriction
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is based on a Boolean expression (called the restriction expression) comparing values of attributes to literals, variables, other attributes, or expressions. The Restrict operator s output relation contains exactly those tuples from the original relation for which the restriction expression evaluates to True. The Restrict operator lters a relation horizontally; in contrast, the Project operator lters a relation vertically. The Project operator is much simpler: in addition to a relation, the Project operator takes, as input, a list of attributes needed for the resulting relation. Note that the proper projection should include unique tuples only; otherwise, the result is not a relation. Nevertheless, RDBMS do not enforce this rule because it is more practical to allow a multiset (or a bag) as the result to send it directly to a client application or to store it temporarily. Figure 3-1 shows the Restrict and the Project operators graphically. Imagine that the right rectangle showing the Project operator represents the relation Employees, with attributes ID, Name, HireDate, DepartmentID, and BirthDate. The Project operator returns a relation with ID, HireDate, and DepartmentID as its attributes, and these attributes are indicated by the darker shading in the gure.
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