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Understanding Relational Databases
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Operators of Relational Algebra
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In previous sections of this chapter, you have studied the terminology and integrity rules of relational databases with the goal of understanding existing relational databases In particu lar, understanding connections among tables was emphasized as a prerequisite to retrieving useful information This section describes some fundamental operators that can be used to retrieve useful information from a relational database You can think of relational algebra similarly to the algebra of numbers except that the objects are different: algebra applies to numbers and relational algebra applies to tables In algebra, each operator transforms one or more numbers into another number Similarly, each operator of relational algebra transforms a table (or two tables) into a new table This section emphasizes the study of each relational algebra operator in isolation For each operator, you should understand its purpose and inputs While it is possible to com bine operators to make complicated formulas, this level of understanding is not important for developing query formulation skills Using relational algebra by itself to write queries can be awkward because of details such as ordering of operations and parentheses There fore, you should seek only to understand the meaning of each operator, not how to combine operators to write expressions The coverage of relational algebra groups the operators into three categories The most widely used operators (restrict, project, and join) are presented first The extended cross product operator is also presented to provide background for the join operator Knowledge of these operators will help you to formulate a large percentage of queries More special ized operators are covered in latter parts of the section The more specialized operators include the traditional set operators (union, intersection, and difference) and advanced operators (summarize and divide) Knowledge of these operators will help you formulate more difficult queries
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restrict
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an operator that re trieves a subset of the rows of the input table that satisfy a given condition
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Restrict (Select) and Project Operators
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The restrict (also known as select) and project operators produce subsets of a table Because users often want to see a subset rather than an entire table, these operators are widely used These operators are also popular because they are easy to understand The restrict and project operators produce an output table that is a subset of an input table (Figure 33) Restrict produces a subset of the rows, while project produces a subset of
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FIGURE 33 Graphical Representation of Restrict and Project Operators
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In this book, the operator name restrict is used to avoid confusion with the SQL SELECT statement The operator is more widely known as select
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3
The Relational Data Model
TABLE 38
OfferNo 3333 5678
Result of Restrict Operation on the Sample Offering Table (TABLE 34)
CourseNo IS320 IS480 OffTerm SPRING SPRING OffYear 2006 2006 OffLocation BLM214 BLM302 OffTime 8:30 AM 10:30 AM FacSSN 098-76-5432 987-65-4321 OffDays MW MW
TABLE 39 Result of a Project Operation on
Offering, CourseNo
CourseNo IS320 IS460 IS480
project an operator that retrieves a specified subset of the columns of the input table
the columns Restrict uses a condition or logical expression to indicate what rows should be retained in the output Project uses a list of column names to indicate what columns to retain in the output Restrict and project are often used together because tables can have many rows and columns It is rare that a user wants to see all rows and columns The logical expression used in the restrict operator can include comparisons involving columns and constants Complex logical expressions can be formed using the logical oper ators AND, OR, and NOT For example, Table 38 shows the result of a restrict operation on Table 34 where the logical expression is: OffDays = ' M W AND OffTerm = 'SPRING' AND OffYear = 2006 A project operation can have a side effect Sometimes after a subset of columns is retrieved, there are duplicate rows When this occurs, the project operator removes the duplicate rows For example, if Offering CourseNo is the only column used in a project operation, only three rows are in the result (Table 39) even though the Offering table (Table 34) has nine rows The column Offering CourseNo contains only three unique values in Table 34 Note that if the primary key or a candidate key is included in the list of columns, the resulting table has no duplicates For example, if OfferNo was included in the list of columns, the result table would have nine rows with no duplicate removal necessary This side effect is due to the mathematical nature of relational algebra In relational al gebra, tables are considered sets Because sets do not have duplicates, duplicate removal is a possible side effect of the project operator Commercial languages such as SQL usually take a more pragmatic view Because duplicate removal can be computationally expensive, duplicates are not removed unless the user specifically requests it
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