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Silberschatz Korth Sudarshan: Database System Concepts, Fourth Edition
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I Data Models
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3 Relational Model
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The McGraw Hill Companies, 2001
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The Relational Algebra
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balance 900 Figure 318 Largest account balance in the bank
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We can now write the temporary relation that consists of the balances that are not the largest: accountbalance ( accountbalance
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< dbalance
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(account d (account)))
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This expression gives those balances in the account relation for which a larger balance appears somewhere in the account relation (renamed as d) The result contains all balances except the largest one Figure 317 shows this relation Step 2: The query to nd the largest account balance in the bank can be written as: balance (account) accountbalance ( accountbalance (account d (account)))
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< dbalance
Figure 318 shows the result of this query As one more example of the rename operation, consider the query Find the names of all customers who live on the same street and in the same city as Smith We can obtain Smith s street and city by writing customer -street, customer -city ( customer -name = Smith (customer )) However, in order to nd other customers with this street and city, we must reference the customer relation a second time In the following query, we use the rename operation on the preceding expression to give its result the name smith-addr, and to rename its attributes to street and city, instead of customer-street and customer-city: customer customer -name ( customer customer -street =smith-addr street customer customer -city=smith-addr city (customer smith-addr (street,city) ( customer -street, customer -city ( customer -name = Smith (customer ))))) The result of this query, when we apply it to the customer relation of Figure 34, appears in Figure 319 The rename operation is not strictly required, since it is possible to use a positional notation for attributes We can name attributes of a relation implicitly by using a positional notation, where $1, $2, refer to the rst attribute, the second attribute, and so on The positional notation also applies to results of relational-algebra operations customer-name Curry Smith Figure 319 Customers who live on the same street and in the same city as Smith
Silberschatz Korth Sudarshan: Database System Concepts, Fourth Edition
I Data Models
3 Relational Model
The McGraw Hill Companies, 2001
3
Relational Model
The following relational-algebra expression illustrates the use of positional notation with the unary operator : $2=$3 (R R) If a binary operation needs to distinguish between its two operand relations, a similar positional notation can be used for relation names as well For example, $R1 could refer to the rst operand, and $R2 could refer to the second operand However, the positional notation is inconvenient for humans, since the position of the attribute is a number, rather than an easy-to-remember attribute name Hence, we do not use the positional notation in this textbook
322 Formal De nition of the Relational Algebra
The operations in Section 321 allow us to give a complete de nition of an expression in the relational algebra A basic expression in the relational algebra consists of either one of the following: A relation in the database A constant relation A constant relation is written by listing its tuples within { }, for example { (A-101, Downtown, 500) (A-215, Mianus, 700) } A general expression in relational algebra is constructed out of smaller subexpressions Let E1 and E2 be relational-algebra expressions Then, these are all relationalalgebra expressions: E1 E2 E1 E2 E1 E2 P (E1 ), where P is a predicate on attributes in E1 S (E1 ), where S is a list consisting of some of the attributes in E1 x (E1 ), where x is the new name for the result of E1
323 Additional Operations
The fundamental operations of the relational algebra are suf cient to express any relational-algebra query1 However, if we restrict ourselves to just the fundamental operations, certain common queries are lengthy to express Therefore, we de ne additional operations that do not add any power to the algebra, but simplify common queries For each new operation, we give an equivalent expression that uses only the fundamental operations
1 In Section 33, we introduce operations that extend the power of the relational algebra, to handle null and aggregate values
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