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พีชคณิตแบบสัมพันธ์ (Relational Algebra) บทที่ 3 อ. ดร. ชุรี เตชะวุฒิ CS (204)321 ระบบฐานข้อมูล 1 (Database System I)

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Outlines 1) Unary relational operations: SELECT and PROJECT 2) Relational algebra operations from set theory 3) Binary relational operations: JOIN and DIVISION 4) Additional relational operations 5) Examples of queries in relational algebra

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Unary relational operations : SELECT The SELECT operation is used to select a subset of the tuples from a relation that satisfy a selection condition. Sigma σ is used to denote the SELECT operator, and the selection condition is a Boolean expression specified on the attributes of relation R. The result of relational operation is a relation. The number of tuples in the resulting relation is always less than or equal to the number of tuples in R. The degree of the relation resulting from a SELECT operation is the same as that of R.

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Unary relational operations : SELECT Clauses of selection condition: is one of the operators Boolean operators AND, OR, NOT can also be used to form a selection condition

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Unary relational operations : SELECT EX1: select the EMPLOYEE tuples whose department is 4. EX2: select the EMPLOYEE tuples whose salary is greater than $30,000. EX3: select the tuples for all employees who either work in department 4 and make over $25,000 per year, or work in department 5 and make over $30,000. (The result is shown in Figure 7.8(a) )

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Unary relational operations : PROJECT The PROJECT operation is used to project the relation over certain attributes of a relation. Pi π is used to present the PROJECT operator. The result of the PROJECT operation has only the attributes specified in and in the same order as they appear in the list. The degree of the relation that is equal to the number of attributes in. Duplicate elimination if the attribute list includes only nonkey attributes of R.

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Unary relational operations : PROJECT EX4: list each employee’s last and first name and salary. (The result is shown in Figure 7.8(b) ) EX5: list each employee’s sex and salary. (The result is shown in Figure 7.8(c) )

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Sequences of operations and RENAME operation Two ways to write the relational algebra: (1) Write single relational algebra expression by nesting the operations, or (2) Apply one operation at a time and create intermediate result relations. EX6: to retrieve the first name, last name, and salary of all employees who work in department number 5. -------- (2) -------- (1) (The result is shown in Figure 7.9(a) )

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Sequences of operations and RENAME operation RENAME operation can be used to rename the attributes in the intermediate and result relations. EX7: from EX1 with RENAME operation (The result is shown in Figure 7.9(b) )

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Sequences of operations and RENAME operation Rho ρ is used to present the RENAME operator. -------- (1) -------- (2) -------- (3) (1) Rename both resulting relation name and attribute names (2) Rename only resulting relation name (3) Rename only attribute names

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Relational algebra operations from set theory Use standard mathematical on sets and on two union-compatible relations R and S UNION INTERSECTION MINUS or DIFFERENCE Combine tuples in R and S and then eliminate duplicate tuples. Include all tuples that are in both R and S. Include all tuples that are in R but not in S.

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Relational algebra operations from set theory Rules for using these operators: Commutative operations Note: DIFFERENCE operation is not commutative Associative operations

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Relational algebra operations from set theory Another set theoretic operation is CARTESIAN PRODUCT or CROSS PRODUCT. Relations on which it is applied do not have to be union compatible. If R has n R tuples and S has n S tuples, then RxS will have n R * n S tuples.

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Relational algebra operations from set theory The operation is useful when followed by a selection that matches values of attributes coming from the component relations. EX8: to retrieve for each female employee a list of the names of her dependents (The result is shown in Figure 7.12 )

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Binary relational operations: JOIN The JOIN operation, denoted by,is used to combine related tuples from two relations into single tuples. The general form of a JOIN operation on two relations is A general join condition is in the form

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Binary relational operations: JOIN The main difference between CARTESIAN PRODUCT and JOIN: JOIN: Only combinations of tuples satisfying the join condition appear in the result. CARTESIAN PRODUCT: All combinations of tuples are included in the result.

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Binary relational operations: JOIN Three kinds of JOIN operations: THETA JOIN Use such a general join condition with one of the comparison operators. EQUIJOIN Involve with equality comparisons only. The result always have one ore more pair of attributes that have identical values in every tuple. NATURAL JOIN (denoted by *) Basically an EQUIJOIN followed by removal of the superfluous attributes (only one join attribute is kept). The removal requires two join attributes have the same name in both relations. If this is not the case, a renaming operation is applied first.

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Binary relational operations: DIVISION The DIVISION operation is denoted as following

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Binary relational operations: DIVISION EX9: Find the answer A: a1 a2 a3 a4 a1 a3 a2 a3 a4 a1 a2 a3 B: b1 b1 b1 b1 b2 b2 b3 b3 b3 b4 b4 b4 R S A: a1 a2 a3 T B: b1 b4 (As shown in Figure 7.15(b) )

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Additional relational operations: AGGREGATE FUNCTION and GROUPING Cannot be perform with the basic relational algebra operations previously described. (2) Use GROUPING to group tuples in a relation by the value of some of their attributes and then applying an AGGREGATE FUNCTIONS independently to each group. Two ways of applications: (1) Use AGGREGATE FUNCTIONS in the case of, for example, retrieving the average or total salary of all employees or the number of employee tuples Must use some common functions including SUM, AVERAGE, MAXIMUM, MINIMUM,COUNT.

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Additional relational operations: AGGREGATE FUNCTION and GROUPING “Script F”,, is used to present AGGREGATE FUNCTION operation. is a list of attributes of the relation specified in R. is a list ofpairs. are such as SUM, AVERAGE, MAXIMUM, MINIMUM,COUNT. is an attribute of the relation specified in R. The resulting relation has the grouping attributes plus one attribute for each element in the function list.

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Additional relational operations: AGGREGATE FUNCTION and GROUPING EX10: group employee tuples by DNO, and list the number of employees in the department, and their average salary. (The result is shown in Figure 7.16(b) )

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Examples of queries in relational algebra Query1: Retrieve the name and address of all employees who work for the ‘Research’ department. Note: The operation order is SELECT – JOIN - PROJECT Query2: Find the names of employees who work on all the projects controlled by department number 5. Query3: List the names of employees who have at least one dependent.

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Chapter 6 The Relational Algebra Copyright © 2004 Ramez Elmasri and Shamkant Navathe.

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