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Relational Algebra

Indra Budi indra@cs.ui.ac.id. Relational Algebra. What is Relational Algebra?. Relational Algebra is a Procedural Paradigm You Need to Tell What/How to Construct the Result Consists of a Set of Operators Which, When Applied to Relations, Yield Relations (Closed Algebra).

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Relational Algebra

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  1. Indra Budi indra@cs.ui.ac.id Relational Algebra

  2. What is Relational Algebra? • Relational Algebra is a Procedural ParadigmYou Need to Tell What/How to Construct the Result • Consists of a Set of Operators Which, When Applied to Relations, Yield Relations (Closed Algebra) • Basic Relational Operations: • Unary Operations • SELECT s • PROJECT  or P. • Binary Operations • Set operations: • UNION  • INTERSECTION  • DIFFERENCE – • CARTESIAN PRODUCT  • JOIN operations 

  3. Relational Algebra R S union R S intersection R \ S set difference R S Cartesian product A1, A2, ..., An (R) projection F (R) selection R S natural join R S theta-join RSdivision [A1 B1,.., An Bn] rename

  4. Selection • Selects the Set of Tuples (Rows) From a Relation, Which Satisfy a Selection Condition • General Form s<selection condition> (R) • R is a Relation • Selection condition is a Boolean Expression on the Attributes of R • Resulting Relation Has the Same Schema as R • Select Finds and Retrieves All Relevant Rows (Tuples) of Table/Relation R which Includes ALL of its Columns (Attributes)

  5. EMP TITLE='Elect. Eng.'(EMP) ENO ENAME TITLE ENO ENAME TITLE E1 J. Doe Elect. Eng. E2 M. Smith Syst. Anal. E3 A. Lee Mech. Eng. E4 J. Miller Programmer E5 B. Casey Syst. Anal. E6 L. Chu Elect. Eng. E7 R. Davis Mech. Eng. E8 J. Jones Syst. Anal. E1 J. Doe Elect. Eng E6 L. Chu Elect. Eng. Selection Example TITLE='Elect. Eng.’ OR TITLE=‘Mech.Eng’(EMP)

  6. Another Selection Example A S C null W B null null

  7. A SELECT Condition is a Boolean Expression Form F1Y F2Y ..., Y Fq (Q>=1), Where Fi (I=1,…,q) are Atomic Boolean Expressions of the Form aqc or aqb, a, b are Attributes of R and c is a Constant. The Operator Q is one of the Arithmetic Comparison Operators: <, >, =, <>, >=, <= The Operator Y is one of the Logical Operators: , , ¬ Nesting: ( ) Selection Condition

  8. Select certain Columns (Attributes) Specified in an Attribute ListX From a Relation R General Form <attribute-list>(R) R is a Relation Attribute-list is a Subset of the Attributes of R Over Which the Projection is Performed Project Retrieves Specified Columns of Table/Relation R which Includes ALL of its Rows (Tuples) Projection

  9. PROJ PNO PNAME BUDGET P1 Instrumentation 150000 P2 Database Develop. 135000 P3 CAD/CAM 250000 P4 Maintenance 310000 P5 CAD/CAM 500000 PNO BUDGET P1 150000 P2 135000 P3 250000 P4 310000 P5 500000 Projection Example PNO,BUDGET(PROJ)

  10. Other Projection Examples

  11. Several Operations can be Combined to form a Relational Algebra Expression (query) Example: Retrieve all Customers over age 60? Method 1:CNAME, ADDRESS, AGE (sAGE>60(CUSTOMER) ) Method 2: Senior-CUST(C#, Addr, Age) = CNAME, ADDRESS, AGE (sAGE>60(CUSTOMER) ) Method 3: CNAME, ADDRESS, AGE (C) where C = sAGE>60(CUSTOMER) Relational Algebra Expression

  12. EMP ENO ENAME TITLE E1 J. Doe Elect. Eng. E2 M. Smith Syst. Anal. E3 A. Lee Mech. Eng. E4 J. Miller Programmer E5 B. Casey Syst. Anal. TITLE(PROJ) E6 L. Chu Elect. Eng. TITLE E7 R. Davis Mech. Eng. Elect.Eng Syst.Anal Mec.Eng Programmer E8 J. Jones Syst. Anal. Characteristics of Projection • The PROJECT Operation Eliminates Duplicate Tuples in the Resulting Relation • Why? • Projection Must Maintain a Mathematical Set (No Duplicate Elements)

  13. Selection with Projection Example

  14. Renaming • The RENAME operator gives a new schema to a relation. • R1 := RENAMER1(A1,…,An)(R2) makes R1 be a relation with attributes A1,…,An and the same tuples as R2. • Simplified notation: R1(A1,…,An) := R2. • Other use () notation • R2  R1(A1,…,An)

  15. Sequences of Assignments • Create temporary relation names. • Renaming can be implied by giving relations a list of attributes. • Example: R3 := R1 JOINC R2 can be written: R4 := R1 * R2 R3 := SELECTC (R4) OR R4  R1 * R2 R3  SELECTC (R4)

  16. General Form RS where R, S are Relations Result contains Tuples from both R and S Duplications are Removed The two Operands R, S should be union-compatible Example: “find students registered for course C1 or C3” s#(CNO=‘C1’ (S-C)) s#(CNO=‘C3’ (S-C)) Union

  17. Two Relations R1(A1, A2, ..., An) and R2(B1, B2, ..., Bn) are said Union-compatible If and Only If They Have The Same Number of Attributes The Domains of Corresponding Attributes are Compatible, i.e., Dom(Ai)=dom(Bi) for I=1, 2, ..., N Names Do Not Have to be Same! For Relational Union and Difference Operations, the Operand Relations Must Be Union Compatible The Resulting Relation for Relational Set Operations Has the Same Attribute Names as the First Operand Relation R1 (by Convention) Union Compatibility

  18. General Form R – S where R and S are Relations Result Contains all tuples that are in R, but not in S. R – S <> S – R Again, there Must be Compatibility Example “Find the students who registered course C1 but not C3”  s#(CNO=‘C1’ (S-C)) –s#(CNO=‘C3’ (S-C)) Set Difference

  19. General Form RS where R and S are Relations Result Contains all Tuples that are in R and S. RS = R – (R – S) Again, there Must be Compatibility Example “find the students who registered for both C1 and C3” s#(CNO=‘C1’ (S-C)) s#(CNO=‘C3’ (S-C)) Set Intersection

  20. Union, Difference, Intersection Examples What are these Other Three Result Tables?

  21. Given Relations R of Degree k1 and Cardinality card1 S of Degree k2 and Cardinality card2 Cartesian Product RS is a Relation of Degree (k1+ k2) and Consists of Tuples of Degree (k1+ k2) where each Tuple is a Concatenation of one Tuple of R with one Tuple of S Cardinality of the Result of the Cartesian Product RS is card1 * card2 What is One Problem with Cartesian Product w.r.t. the Result Set? Cartesian Product

  22. Cartesian Product: Example R S A B C E F a1 a2 a3 b1 b1b4 c3 c5 c7 e1 e2 f1 f5 A B C E F R S  a1 a1 a2 a2 a3 a3 b1 b1 b1 b1 b4 b4 c3 c3 c5 c5 c7 c7 e1 e2 e1 e2 e1 e2 f1 f5 f1 f5 f1 f5

  23. Given R(A1, …,An) and S(B1,…,Bm), the result of a Cartesian product RS is a relation of schema R’(A1, …, An, B1, …, Bm). Example “Get a list containing (S#, C#) for all students who live in Storrs but are not registered for the database course” Cartesian Product (S#(city=‘Storrs’(STUDENT))   C# (CNAME=‘Database’(COURSE))) –S#, C#(S-C)

  24. EMP SAL EMP ENO ENAME TITLE ENO ENAME EMP.TITLE SAL.TITLE SAL E1 J. Doe Elect. Eng E1 J. Doe Elect. Eng. Elect. Eng. 40000 E2 M. Smith Syst. Anal. E1 J. Doe Elect. Eng. Syst. Anal. 34000 E3 A. Lee Mech. Eng. E1 J. Doe Elect. Eng. Mech. Eng. 27000 E4 J. Miller Programmer E1 J. Doe Elect. Eng. Programmer 24000 E5 B. Casey Syst. Anal. E2 M. Smith Syst. Anal. Elect. Eng. 40000 E6 L. Chu Elect. Eng. E2 M. Smith Syst. Anal. Syst. Anal. 34000 E7 R. Davis Mech. Eng. E2 M. Smith Syst. Anal. Mech. Eng. 27000 E8 J. Jones Syst. Anal. E2 M. Smith Syst. Anal. Programmer 24000 E3 A. Lee Mech. Eng. Elect. Eng. 40000 E3 A. Lee Mech. Eng. Syst. Anal. 34000 SAL E3 A. Lee Mech. Eng. Mech. Eng. 27000 E3 A. Lee Mech. Eng. Programmer 24000 TITLE SAL Elect. Eng. 40000 Syst. Anal. 34000 E8 J. Jones Syst. Anal. Elect. Eng. 40000 Mech. Eng. 27000 E8 J. Jones Syst. Anal. Syst. Anal. 34000 Programmer 24000 E8 J. Jones Syst. Anal. Mech. Eng. 27000 E8 J. Jones Syst. Anal. Programmer 24000 Cartesian Product Example

  25. The Join Operation • Used to combine related tuples from two relation into single tuples with some condition • Cartesian product all combination of tuples are included in the result, meanwhile in the join operation, only combination of tuples satisfying the join condition appear in the result

  26. General Form RS where R, S are Relations,  is a Boolean Expression, called a Join Condition. A Derivative of Cartesian Product R S =  (RS) R(A1, A2, ..., Am, B1, B2, ..., Bn) is the Resulting Schema of a -Join over R1 and R2: R1(A1, A2, ..., Am)  R2 (B1, B2, ..., Bn) Theta Join (-Join)

  27. A -Join Condition is a Boolean Expression of the form F1y1F2y2 ..., yn-1Fq (q>=1), where Fi (i=1,…,q) are Atomic Boolean Expressions of the form Ai  Bj, Ai, Bj are Attributes of R1 and R2 Respectively q is one of the Algorithmic Comparison Operators =, <>, >, <. >=, <= The Operator yi (i=1,…,n-1) is Either a Logical AND operator  or a logical OR operator  -Join Condition

  28. EMP EMP E.TITLE=SAL.TITLE SAL ENO ENAME TITLE SAL.TITLE TITLE SAL E1 J. Doe Elect. Eng E2 M. Smith Syst. Anal. Elect. Eng. Elect. Eng. 40000 E3 A. Lee Mech. Eng. Analyst Analyst 34000 E4 J. Miller Programmer Mech. Eng. Mech. Eng. 27000 E5 B. Casey Syst. Anal. 24000 Programmer Programmer E6 L. Chu Elect. Eng. Syst. Anal. Syst. Anal. 34000 E7 R. Davis Mech. Eng. Elect. Eng. Elect. Eng. 40000 E8 J. Jones Syst. Anal. Mech. Eng. Mech. Eng. 27000 Syst. Anal. Syst. Anal. 34000 SAL TITLE SAL Elect. Eng. 40000 Syst. Anal. 34000 Mech. Eng. 27000 Programmer 24000 -Join Example ENO ENAME E1 J. Doe E2 M. Smith E3 A. Lee E4 J. Miller E5 B. Casey E6 L. Chu E7 R. Davis E8 J. Jones

  29. Equi-join (EQUIJOIN) The  Expression only Contains one or more Equality Comparisons Involving Attributes from R1 and R2 Natural Join Denoted as R S Special Equi-join of Two Relations R and S Over a Set of Attributes Common to both R and S By Common, it means that each Join Attribute in A has not only Compatible Domains but also the SameName in both Relations R and S Other Types of Join

  30. R R.B=S.B S R S Examples R S A B C B E a1 a2 a3 b1 b1b4 c3 c5 c7 b1 b5 e1 e2 EQUIJOIN Natural Join A R.B C E A R.B S.B C E a1 a2 b1 b1 c3 c5 e1 e1 a1 a2 b1 b1 b1 b1 c3 c5 e1 e1

  31. Natural Join Combines Relations on Attributes with the Same Names STUDENT(S#, SN, CITY, Email) S-C(S#, C#, G) Example Query 1: “list of students with complete course grade info” STUDENT S-C All Natural Joins can be Expressed by a Combination of Primitive Operators Example Query 2: “print all students info (courses taken and grades)” S#, SN, CITY, Email, C#, G (STUDENT.S# = S-C.S# (STUDENT S-C)) Natural Join

  32. EMP ENO ENAME TITLE E1 J. Doe Elect. Eng E2 M. Smith Syst. Anal. E3 A. Lee Mech. Eng. E4 J. Miller Programmer E5 B. Casey Syst. Anal. E6 L. Chu Elect. Eng. E7 R. Davis Mech. Eng. E8 J. Jones Syst. Anal. SAL TITLE SAL Elect. Eng. 70000 Syst. Anal. 80000 Mech. Eng. 56000 Programmer 60000 Natural Join Example EMP SAL SAL ENO ENAME E.TITLE E1 J. Doe Elect. Eng. 70000 E2 M. Smith Syst. Anal. 80000 E3 A. Lee Mech. Eng. 56000 E4 J. Miller Programmer 60000 E5 B.Casey Syst.Anal 80000 E6 L. Chu Elect.Eng 70000 E7 R.Davis Mech.Eng 56000 E8 J. Jones Syst. Anal. 80000

  33. Another Natural Join Example

  34. 1 4 5 5 Yet Another Natural Join Example

  35. Quotient (Division) • Given Relations • R(T,U) of degree r • S(U) of degree s • The Division of R by S, R ÷ S • Results is a Relation of Degree (rs) • Consists of all (rs)-tuples t such that for all s-tuples u in S, the tuple tu is in R.

  36. R ENO PNO PNAME BUDGET 150000 E1 P1 Instrumentation E2 P1 Instrumentation 150000 E2 P2 Database Develop. 135000 E3 P1 Instrumentation 150000 E3 P4 Maintenance E4 P2 Instrumentation E5 P2 Instrumentation E6 P4 Maintenance E7 P3 CAD/CAM 310000 E8 P3 CAD/CAM 150000 S 150000 310000 250000 250000 PNAME PNO BUDGET P1 Instrumentation 150000 P4 Maintenance 310000 Division Example Find the employees who work for both project P1 and project P4? R ÷ S ENO E3

  37. “list the S# of students that have taken all of the courses listed in S-C” S# (S-C)C# (S-C) “list the S# of students who have taken all of the courses taught by instructor Smith” S# (S-CC# (Instructor=‘Smith’(C))) Division: Another Example

  38. Relational Algebra • Selection • Projection • Union • Difference • Cartesian Product • Intersection • Join, Equi-join, Natural Join Derivable from the fundamental operators Fundamental Operators

  39. A Set of Relational Algebra Operations Is Called a Complete Set, If and Only If Any Relational Algebra Operator in the Set Cannot be Derived in Terms of a Sequence of Others in Set Any Relational Algebra Operator Not in the Set Can Be Derived in Terms of a Sequence of Only the Operators in the Set Important Concepts: The Set of Algebra Operations {S ,P , , –, } is a Complete Set of Relational Algebra Operations Any Query Language Equivalent to These Five Operations is Called Relationally Complete All Relational Algebra Operations

  40. Fundamental Operators Selection Projection Union Set Difference Cartesian Product Additional Operators Join Intersection Quotient (Division) Union Compatibility Same Degree Corresponding Attributes Defined Over the Same Domain Relational Algebra: Summary Form: <Operator><Operand(s)> Result>  Relation (s) Relation

  41. References • Elmasri & Navathe, “Fundamental of Database Systems 3rd ed”, Addison-Wesley, 2000

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