relational algebra indra budi [email protected]. fakultas ilmu komputer ui 2 n basic relational...

Relational Algebra

Indra [email protected]

2 Fakultas Ilmu Komputer UI

Basic Relational Operations: Unary Operations

SELECT PROJECT or .

Binary Operations Set operations:

UNION INTERSECTION DIFFERENCE –

CARTESIAN PRODUCT JOIN operations

What is Relational Algebra?What is Relational Algebra?

Relational Algebra is a Procedural ParadigmYou Need to Tell What/How to Construct the ResultConsists of a Set of Operators Which, When Applied to Relations, Yield Relations (Closed Algebra)


Relational Algebra

R S unionR S intersectionR \ S set differenceR S Cartesian

product

A1, A2, ..., An (R) projection

F (R) selection

R S natural join

R S theta-joinRS division[A1 B1,.., An Bn] rename


SelectionSelection

Selects the Set of Tuples (Rows) From a Relation, Which Satisfy a Selection ConditionGeneral Form selection condition> (R)

R is a Relation Selection condition is a Boolean Expression on the Attributes of RResulting 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)


ENO ENAME TITLE

E1 J. Doe Elect. EngE6 L. Chu Elect. Eng.

TITLE='Elect. Eng.'(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.

EMP

TITLE='Elect. Eng.’ OR TITLE=‘Mech.Eng’(EMP)

Selection ExampleSelection Example


Another Selection ExampleAnother Selection Example

ASCnullWBnullnull


A SELECT Condition is a Boolean Expression

Form F1 F2 Fq (Q>=1), Where

Fi (I=1,…,q) are Atomic Boolean Expressions of the Form

ac or ab, a, b are Attributes of R and c is a Constant.

The Operator is one of the Arithmetic Comparison Operators: <, >, =, <>The Operator is one of the Logical Operators: , , ¬Nesting: ( )

Selection Condition


Select certain Columns (Attributes) Specified in an Attribute List X From a Relation R

General Form <attribute-list>(R)R is a RelationAttribute-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


Projection Example

PNO,BUDGET(PROJ)

PNO BUDGET

P1 150000

P2 135000

P3 250000P4 310000P5 500000

PROJ

PNO BUDGET

P2 135000

P3 250000

P4 310000P5 500000

PNAME

P1 150000Instrumentation

Database Develop.

CAD/CAM

MaintenanceCAD/CAM


Other Projection Examples


Relational Algebra ExpressionRelational Algebra Expression

Several Operations can be Combined to form a Relational Algebra Expression (query)Example: Retrieve all Customers over age 60?

Method 1: CNAME, ADDRESS, AGE ( AGE>60(CUSTOMER) )

Method 2: Senior-CUST(C#, Addr, Age)

= CNAME, ADDRESS, AGE ( AGE>60(CUSTOMER) )

Method 3: CNAME, ADDRESS, AGE (C) where

C AGE>60(CUSTOMER)


ENO ENAME TITLE









EMP TITLE(PROJ)

Elect.EngSyst.AnalMec.EngProgrammer

TITLE

Characteristics of ProjectionCharacteristics of Projection

The PROJECT Operation Eliminates Duplicate Tuples in the Resulting Relation

Why?Projection Must Maintain a Mathematical Set (No Duplicate Elements)


Selection with Projection Example


RenamingRenaming

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)


Sequences of AssignmentsSequences 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 * R2R3 := SELECTC (R4)

ORR4 R1 * R2R3 SELECTC (R4)


General FormR S

where R, S are Relations Result contains Tuples from both R and SDuplications are RemovedThe 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


Union CompatibilityUnion Compatibility

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 AttributesThe Domains of Corresponding Attributes are Compatible, i.e., Dom(Ai)=dom(Bi) for I=1, 2, ..., NNames Do Not Have to be Same!

For Relational Union and Difference Operations, the Operand Relations Must Be Union CompatibleThe Resulting Relation for Relational Set Operations

Has the Same Attribute Names as the First Operand Relation R1 (by Convention)


General Form

R – S

where R and S are RelationsResult 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 DifferenceSet Difference


General Form

R S

where R and S are RelationsResult 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 IntersectionSet Intersection


Union, Difference, Intersection ExamplesUnion, Difference, Intersection Examples

What are these OtherThree Result Tables?


Given RelationsR 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


Cartesian Product: Example Cartesian Product: Example

A B C

a1a2a3

b1b1b4

c3c5c7

FE

f1f5

e1e2

A B C E

a1a1a2a2a3a3

b1b1b1b1b4b4

c3c3c5c5c7c7

e1e2e1e2e1e2

R S

R S F

f1f5f1f5f1f5


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)


Cartesian Product Example

ENO ENAME EMP.TITLE SAL.TITLE SAL





Elect. Eng. 40000

Syst. Anal. 34000

Mech. Eng. 27000

Programmer 24000





Elect. Eng. 40000

Syst. Anal. 34000

Mech. Eng. 27000

Programmer 24000

Elect. Eng. 40000

Syst. Anal. 34000

Mech. Eng. 27000

Programmer 24000

Elect. Eng. 40000

Syst. Anal. 34000

Mech. Eng. 27000

Programmer 24000









EMP SAL

ENO ENAME TITLE

E1 J. Doe Elect. Eng








EMP

TITLE SAL

SAL

Elect. Eng. 40000

Syst. Anal. 34000

Mech. Eng. 27000

Programmer 24000


The Join OperationThe Join Operation

Used to combine related tuples from two relation into single tuples with some conditionCartesian 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


General FormR S

whereR, S are Relations, is a Boolean Expression, called a Join Condition.

A Derivative of Cartesian ProductR 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)Theta Join (-Join)


A -Join Condition is a Boolean Expression of the form F1 1F2 2n-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

is one of the Algorithmic Comparison Operators =, <>, >, <. >=, <= The Operatori (i=1,…,n-1) is Either a Logical AND operator or a logical OR operator

-Join Condition


-Join Example

ENO ENAME TITLE









EMP

TITLE SAL

SAL

Elect. Eng. 40000

Syst. Anal. 34000

Mech. Eng. 27000

Programmer 24000

ENO ENAME

E1 J. Doe

M. SmithE2

E3 A. Lee

E4 J. Miller

E5 B. Casey

E6 L. Chu

E7 R. DavisE8 J. Jones

TITLE

Elect. Eng.

Analyst

Mech. Eng.

Programmer

Syst. Anal.

Elect. Eng.

Mech. Eng.Syst. Anal.

SAL

40000

34000

27000

24000

34000

40000

2700034000

EMP E.TITLE=SAL.TITLE SAL

SAL.TITLE

Elect. Eng.

Analyst

Mech. Eng.

Programmer

Syst. Anal.

Elect. Eng.

Mech. Eng.Syst. Anal.


Other Types of Join

Equi-join (EQUIJOIN)The Expression only Contains one or more Equality Comparisons Involving Attributes from R1 and R2

Natural JoinDenoted as R SSpecial Equi-join of Two Relations R and S Over a Set of Attributes Common to both R and SBy Common, it means that each Join Attribute in A has not only Compatible Domains but also the Same Name in both Relations R and S


ExamplesExamples

A B C

a1a2a3

b1b1b4

c3c5c7

B E

b1b5

e1e2

R S

R R.B=S.B S

A R.B

a1a2

b1b1

C E

c3c5

e1e1

S.B

b1b1

EQUIJOIN

A R.B C E

a1a2

b1b1

c3c5

e1e1

R SNatural Join


Natural Join

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 OperatorsExample 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 Example

ENO ENAME TITLE









EMP

TITLE SAL

SAL

Elect. Eng. 70000

Syst. Anal. 80000

Mech. Eng. 56000

Programmer 60000

ENO ENAME E.TITLE SAL

E1 J. Doe Elect. Eng. 70000

E2 M. Smith Syst. Anal. 80000

56000

80000



EMP SAL

60000E4 J. Miller Programmer

80000E5 B.Casey Syst.Anal

70000E6 L. Chu Elect.Eng

56000E7 R.Davis Mech.Eng


Another Natural Join Example


Yet Another Natural Join Example

1455


Given RelationsR(T,U) of degree rS(U) of degree s

The Division of R by S, R ÷ S

Results is a Relation of Degree (rs) Consists of all (rs)-tuples t such that for

all s-tuples u in S, the tuple tu is in R.

Quotient (Division)


Division Example

ENO

E3

R ÷ S

ENO PNO PNAME

E1 P1 Instrumentation 150000

BUDGET

E2 P1 Instrumentation 150000E2 P2 Database Develop. 135000E3 P1 InstrumentationE3 P4 MaintenanceE4 P2 InstrumentationE5 P2 InstrumentationE6 P4E7 P3 CAD/CAME8 P3 CAD/CAM

310000150000150000310000250000250000

R

Maintenance

150000

S

PNO PNAME BUDGET

P1 Instrumentation 150000P4 Maintenance 310000

Find the employees who work for both project P1 and project P4?


Division: Another Example

“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-CC#

(Instructor=‘Smith’(C)))


Relational Algebra

Fundamental OperatorsDerivable from the fundamental operators

SelectionProjectionUnionDifferenceCartesian Product

Intersection Join, Equi-join, Natural Join


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 SetAny 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 { , , , –, } is a

Complete Set of Relational Algebra Operations

Any Query Language Equivalent to These Five Operations is Called Relationally Complete

All Relational Algebra OperationsAll Relational Algebra Operations


Fundamental Operators

SelectionProjectionUnionSet DifferenceCartesian Product

Relational Algebra: Summary

Form:

<Operator><Operand(s)> Result>

Relation (s) Relation

Additional OperatorsJoinIntersectionQuotient (Division)

Union CompatibilitySame DegreeCorresponding Attributes Defined Over the Same Domain


ReferencesReferences

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

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