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Relational Algebra: Specifying the Operations
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With relational algebra, we describe queries by considering a sequence of operations or manipulations on the tables involved. Some operations act on one table, while others are
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CHAPTER 1 RELA TION AL DA TA BAS E OVERVIEW
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different ways of combining data from two tables. (Remember that when I talk about tables, I really mean ones with unique rows.) Every time we use one of the operations on a table, the result is another table. This means we can build up quite complicated queries by taking the result of one operation and applying another operation to it. We will look at all the different operations in detail throughout the book, but just as a simple example we will discuss how to use relational algebra to retrieve the names of the senior members of our golf club. We will need two operations. The select operation returns just those rows from a table that satisfy a particular condition. The project operation returns just the specified columns. First we ll get just the rows we need. We can say it like this: Apply the select operation to the Member table with the condition that the MemberType field must have the value Senior . Clearly, this is all going to get a bit wordy as we apply more and more operations, so it is useful to introduce some shorthand, as shown in Listing 1-7. (the Greek letter sigma) stands for the select operation, and the condition is specified in the subscript. For convenience I have called the resulting table SenMemb.
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Listing 1-7. The Select Operation to Retrieve the Subset of Rows for Seniors
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SenMemb MemberType= 'Senior' ( Member )
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Figure 1-11 shows the result of this operation. Having retrieved a table with the appropriate rows, we now apply the project operation to get the right columns. Listing 1-8 shows the shorthand for this, where (pi) denotes the project operation and the columns are specified in the subscript.
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Listing 1-8. The Project Operation to Retrieve a Subset of Columns
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Final LastName, FirstName (SenMemb )
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You can express the whole algebra expression in one go, as shown in Listing 1-9.
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Listing 1-9. The Complete Algebra Expression
Final LastName, FirstName ( MemberType= 'Senior' (Member))
Figure 1-11 shows the original, intermediate, and final tables. Note that the intermediate and final tables are not permanent in the database. The example in Figure 1-11 shows how we can apply two relational algebra operations in succession to retrieve a final relation with the required data. We do not really need the power of the relational algebra to visualize how to formulate a query this simple; however, most queries are not this simple.
CHAPTER 1 RELATION AL DATABA SE OV ERVIEW
MemberType= Senior (Member)
LastName, FirstName
MemberType= Senior (Member))
Figure 1-11. Result of two successive relational algebra operations
Relational Calculus: Specifying the Result
Relational algebra lets us specify a sequence of operations that eventually result in a set of rows with the information we require. As we will see throughout this book, there may be several different ways of applying a sequence of relational operations that will retrieve the same data. The other method that relational theory provides for describing a query is relational calculus. Rather than specifying how to do the query, we describe what conditions the resulting data should satisfy. Once again, this may take a bit of getting used to, so we will go over all this more carefully in later chapters. In nonformal language, a relational calculus description of a query has the following form: I want the set of rows that obey the following conditions . . . As with the algebra version, this can become very wordy, so shorthand is convenient, as shown in Listing 1-10.
Listing 1-10. General Form of a Query Expressed in Relational Calculus { m | condition(m) }
The part on the left of the bar will contain a description of the attributes or columns we want returned, while the part on the right describes the criteria they must satisfy. The letter m is a way of referring to a particular row (m) in a table, and we will need to introduce other labels when we have several tables to contend with. An example is the best way to clarify what a relational calculus expression means. Listing 1-11 shows the relational calculus for the query to retrieve senior club members.
Listing 1-11. Relational Calculus to Retrieve Senior Members {m | Member(m) and m.MemberType = 'Senior'}