rel algebra

8/6/2019 Rel Algebra

1/58

Chapter 6

The Relational Algebra


2/58

Slide 6- 2

Chapter Outline

Relational Algebra

Unary Relational Operations

Relational Algebra Operations From Set Theory

Binary Relational Operations Additional Relational Operations

Examples of Queries in Relational Algebra

Example Database Application (COMPANY)


3/58

Slide 6- 3

Relational Algebra Overview

Relational algebra is the basic set of operationsfor the relational model

These operations enable a user to specify basicretrieval requests (orqueries)

The result of an operation is a new relation, whichmay have been formed from one or more inputrelations

This property makes the algebra closed (allobjects in relational algebra are relations)


4/58

Slide 6- 4

Relational Algebra Overview (continued)

The algebra operations thus produce newrelations

These can be further manipulated using

operations of the same algebra

A sequence of relational algebra operationsforms a relational algebra expression

The result of a relational algebra expression is also arelation that represents the result of a databasequery (or retrieval request)


5/58

Slide 6- 5

Brief History of Origins of Algebra(from the author Navathe)

Muhammad ibn Musa al-Khwarizmi (800-847 CE) wrote abook titled al-jabr about arithmetic of variables Book was translated into Latin. Its title (al-jabr) gave Algebra its name.

Al-Khwarizmi called variables shay Shay is Arabic for thing. Spanish transliterated shay as xay (x was sh in Spain). In time this word was abbreviated as x.

Where does the word Algorithm come from?

Algorithm originates from al-Khwarizmi" Reference: PBS (http://www.pbs.org/empires/islam/innoalgebra.html)


6/58

Slide 6- 6

Relational Algebra Overview

Relational Algebra consists of several groups of operations Unary Relational Operations

SELECT (symbol: W (sigma)) PROJECT (symbol: T (pi)) R

ENAME (symbol: V (rho)) Relational Algebra Operations From Set Theory

UNION ( ), INTERSECTION ( ), DIFFERENCE (or MINUS, ) CARTESIAN PRODUCT ( x )

Binary Relational Operations JOIN (several variations of JOIN exist)

DIVISION Additional Relational Operations

OUTER JOINS, OUTER UNION AGGREGATE FUNCTIONS (These compute summary of

information: for example, SUM, COUNT, AVG, MIN, MAX)


7/58

Slide 6- 7

Database State for COMPANY

All examples discussed below refer to the COMPANY databaseshown here.


8/58

Slide 6- 8

Unary Relational Operations: SELECT

The SELECT operation (denoted by W (sigma)) is used to select asubsetof the tuples from a relation based on a selection condition.

The selection condition acts as a filter

Keeps only those tuples that satisfy the qualifying condition

Tuples satisfying the condition are selectedwhereas theother tuples are discarded (filtered out)

Examples:

Select the EMPLOYEE tuples whose department number is 4:

W DNO = 4 (EMPLOYEE) Select the employee tuples whose salary is greater than $30,000:

W SALARY > 30,000 (EMPLOYEE)


9/58

Slide 6- 9

Unary Relational Operations: SELECT

In general, the selectoperation is denoted by

W (R) where

the symbol W (sigma) is used to denote the selectoperator

the selection condition is a Boolean (conditional)expression specified on the attributes of relation R

tuples that make the condition true are selected appear in the result of the operation

tuples that make the condition false are filtered out discarded from the result of the operation


10/58

Can use Comparison Operators

The Select OperatorThe Select Operator ---- WW

! { " u e

Selection predicate may includecomparison between two attributes

Wsalary < 30000(Employee)

Find those employee having salary less than 30000


11/58

The Select OperatorThe Select Operator ---- WW

Can also use ConnectivesAND

OR

Wdno=5AND salary < 30000(Employee)

Find those employee of department number 5 and salaryless than 30000


12/58

Slide 6- 12

Unary Relational Operations: SELECT(contd.)

SELECT Operation Properties The SELECT operation W (R) produces a relation

S that has the same schema (same attributes) as R SELECT W is commutative:

W (W < condition2> (R)) = W (W < condition1> (R)) Because of commutativity property, a cascade (sequence) of

SELECT operations can be applied in any order: W(W (W (R)) = W (W (W ( R)))

A cascade of SELECT operations can be replaced by a singleselection with a conjunction of all the conditions:

W(W< cond2> (W(R)) = W AND < cond2> AND (R)))

The number of tuples in the result of a SELECT is less than(or equal to) the number of tuples in the input relation R


13/58

Slide 6- 13

The following query results refer to thisdatabase state


14/58

Slide 6- 14

Unary Relational Operations: PROJECT

PROJECT Operation is denoted by T (pi)

This operation keeps certain columns (attributes)from a relation and discards the other columns.

PROJECT creates a vertical partitioning The list of specified columns (attributes) is kept in

each tuple

The other attributes in each tuple are discarded

Example: To list each employees first and lastname and salary, the following is used:

TLNAME, FNAME,SALARY(EMPLOYEE)


15/58

Slide 6- 15

Unary Relational Operations: PROJECT(cont.)

The general form of theprojectoperation is:T(R)

T (pi) is the symbol used to represent theprojectoperation is the desired list of attributes from

relation R.

The project operation removes any duplicate

tuples This is because the result of theprojectoperation

must be a set of tuples Mathematical sets do not allowduplicate elements.


16/58

Slide 6- 16

Unary Relational Operations: PROJECT(contd.)

PROJECT Operation Properties

The number of tuples in the result of projectionT(R) is always less or equal to the number of

tuples in R If the list of attributes includes a keyofR, then the

number of tuples in the result of PROJECT is equalto the number of tuples in R

PROJECT is notcommutative T (T (R) ) = T (R) as long as

contains the attributes in


17/58

Slide 6- 17

Examples of applying SELECT andPROJECT operations


18/58

Slide 6- 18

Relational Algebra Expressions

We may want to apply several relational algebraoperations one after the other

Either we can write the operations as a single

relational algebra expression by nesting theoperations, or

We can apply one operation at a time and createintermediate result relations.

In the latter case, we must give names to therelations that hold the intermediate results.


19/58

Slide 6- 19

Single expression versus sequence ofrelational operations (Example)

To retrieve the first name, last name, and salary of allemployees who work in department number 5, we mustapply a select and a project operation

We can write asingle relational algebra expre

ssion asfollows:

TFNAME, LNAME, SALARY(W DNO=5(EMPLOYEE))

OR We can explicitly show the sequence of operations,giving a name to each intermediate relation:

DEP5_EMPSn W DNO=5(EMPLOYEE)

RESULTn T FNAME, LNAME, SALARY (DEP5_EMPS)


20/58

Slide 6- 20

Unary Relational Operations: RENAME

The RENAME operator is denoted by V (rho)

In some cases, we may want to rename theattributes of a relation or the relation name or

both

Useful when a query requires multipleoperations

Necessary in some cases (see JOIN operationlater)


21/58

Slide 6- 21

Unary Relational Operations: RENAME(contd.)

The general RENAME operation V can beexpressed by any of the following forms:

VS (B1, B2, , Bn )(R) changes both:

the relation name to S,and

the column (attribute) names to B1, B2, ..Bn

VS(R) changes:

the relation name only to S

V(B1, B2, , Bn )(R) changes:

the column (attribute) names only to B1, B2, ..Bn


22/58

Slide 6- 22

Unary Relational Operations: RENAME(contd.)

For convenience, we also use a shorthandforrenaming attributes in an intermediate relation:

If we write:

RESULTn T FNAME, LNAME, SALARY (DEP5_EMPS) RESULT will have the same attribute names as

DEP5_EMPS (same attributes as EMPLOYEE)

If we write:

RESULT (F, M, L, S, B, A, SX, SAL, SU, DNO)nV RESULT (F.M.L.S.B,A,SX,SAL,SU, DNO)(DEP5_EMPS)

The 10 attributes of DEP5_EMPS are renamedtoF, M, L, S, B, A, SX, SAL, SU, DNO, respectively


23/58

Slide 6- 23

Example of applying multiple operationsand RENAME


24/58

Slide 6- 24

Relational Algebra Operations fromSet Theory: UNION

UNION Operation

Binary operation, denoted by

The result ofR S, is a relation that includes all

tuples that are either in R or in S or in both R andS

Duplicate tuples are eliminated

The two operand relations R and S must be typecompatible (or UNION compatible)

R and S must have same number of attributes

Each pair of corresponding attributes must be typecompatible (have same or compatible domains)


25/58

Slide 6- 25

Relational Algebra Operations fromSet Theory: UNION

Example: To retrieve the social security numbers of all employees who

eitherwork in department 5(RESULT1 below) ordirectlysupervise an employee who works in department 5(RESULT2below)

We can use the UNION operation as follows:DEP5_EMPS n WDNO=5 (EMPLOYEE)RESULT1n T SSN(DEP5_EMPS)

RESULT2(SSN)n TSUPERSSN(DEP5_EMPS)RESULTn RESULT1 RESULT2

The union operation produces the tuples that are in eitherRESULT1 orRESULT2 or both


26/58

Slide 6- 26

Example of the result of a UNIONoperation

UNION Example


27/58

Slide 6- 27

Relational Algebra Operations fromSet Theory

Type Compatibility of operands is required for the binaryset operation UNION , (also for INTERSECTION , andSET DIFFERENCE , see next slides)

R1(A1, A2, ..., An) and R2(B1, B2, ..., Bn) are typecompatible if:

they have the same number of attributes, and

the domains of corresponding attributes are type compatible(i.e. dom(Ai)=dom(Bi) for i=1, 2, ..., n).

The resulting relation forR1R2 (also forR1R2, orR1R2, see next slides) has the same attribute names as thefirstoperand relation R1 (by convention)


28/58

Slide 6- 28

Relational Algebra Operations from SetTheory: INTERSECTION

INTERSECTION is denoted by

The result of the operation R S, is a

relation that includes all tuples that are inboth R and S

The attribute names in the result will be thesame as the attribute names in R

The two operand relations R and S must betype compatible


29/58

Slide 6- 29

Relational Algebra Operations from SetTheory: SET DIFFERENCE (cont.)

SET DIFFERENCE (also called MINUS orEXCEPT) is denoted by

The result ofR S, is a relation that includes all

tuples that are in R but not in S

The attribute names in the result will be thesame as the attribute names in R

The two operand relations R and S must betype compatible


30/58

Slide 6- 30

Example to illustrate the result of UNION,INTERSECT, and DIFFERENCE


31/58

Slide 6- 31

Some properties of UNION, INTERSECT,and DIFFERENCE

Notice that both union and intersection are commutativeoperations; that is

R S = S R, and R S = S R

Both union and intersection can be treated as n-aryoperations applicable to any number of relations as bothare associative operations; that is

R (S T) = (R S) T

(R S) T = R (S T)

The minus operation is not commutative; that is, ingeneral

R S S R


32/58

Slide 6- 32

Relational Algebra Operations from SetTheory: CARTESIAN PRODUCT

CARTESIAN (or CROSS) PRODUCT Operation

This operation is used to combine tuples from two relationsin a combinatorial fashion.

Denoted by R(A1, A2, . . ., An) x S(B1, B2, . . ., Bm)

Result is a relation Q with degree n + m attributes:

Q(A1, A2, . . ., An, B1, B2, . . ., Bm), in that order.

The resulting relation state has one tuple for eachcombination of tuplesone from R and one from S.

Hence, ifR has nR tuples (denoted as |R| = nR ), and S hasnS tuples, then R x S will have nR * nS tuples.

The two operands do NOT have to be "type compatible


33/58

Slide 6- 33

Relational Algebra Operations from SetTheory: CARTESIAN PRODUCT (cont.)

Generally, CROSS PRODUCT is not ameaningful operation

Can become meaningful when followed by other

operations Example (not meaningful):

FEMALE_EMPS n W SEX=F(EMPLOYEE)

EMPNAMES n T FNAME, LNAME, SSN (FEMALE_EMPS)

EMP_DEPENDENTS n EMPNAMES x DEPENDENT

EMP_DEPENDENTS will contain every combination ofEMPNAMES and DEPENDENT whether or not they are actually related


34/58

Slide 6- 34

Relational Algebra Operations from SetTheory: CARTESIAN PRODUCT (cont.)

To keep only combinations where theDEPENDENT is related to the EMPLOYEE, weadd a SELECT operation as follows

Example (meaningful): FEMALE_EMPS n W SEX=F(EMPLOYEE)

EMPNAMES n T FNAME, LNAME, SSN (FEMALE_EMPS)

EMP_DEPENDENTS n EMPNAMES x DEPENDENT

ACTUAL_DEPS n W SSN=ESSN(EMP_DEPENDENTS)

RESULTn T FNAME, LNAME, DEPENDENT_NAME (ACTUAL_DEPS)

RESULT will now contain the name of female employeesand their dependents


35/58

A Simple Cartesian ProductA Simple Cartesian Product

Two one-attribute relations

a

A

B

C

b

X

Y

Z

A B

a b

A X

A Y

A Z

B X

B Y

B Z

C X

C Y

C Z

Their cartesian product

A X B


36/58

Slide 6- 36

Example of applying CARTESIANPRODUCT


37/58

Slide 6- 37

Binary Relational Operations: JOIN

JOIN Operation (denoted by ) The sequence of CARTESIAN PRODUCT followed by

SELECT is used quite commonly to identify and selectrelated tuples from two relations

A special operation, called JOIN combines this sequenceinto a single operation

This operation is very important for any relational databasewith more than a single relation, because it allows uscombine related tuples from various relations

The general form of a join operation on two relations R(A1,

A2, . . ., An) and S(B1, B2, . . ., Bm) is:R Swhere R and S can be any relations that result from generalrelational algebra expressions.


38/58

Slide 6- 38

Binary Relational Operations: JOIN (cont.)

Example: Suppose that we want to retrieve the name of themanager of each department. To get the managers name, we need to combine each

DEPARTMENT tuple with the EMPLOYEE tuple whose SSNvalue matches the MGRSSN value in the department tuple.

We do this by using the join operation.

DEPT_MGRn DEPARTMENT MGRSSN=SSN EMPLOYEE

MGRSSN=SSN is the join condition Combines each department record with the employee who

manages the department The join condition can also be specified as

DEPARTMENT.MGRSSN= EMPLOYEE.SSN


39/58

Slide 6- 39

Example of applying the JOIN operation

DEPT_MGRn DEPARTMENT MGRSSN=SSN EMPLOYEE


40/58

Slide 6- 40

Some properties of JOIN

Consider the following JOIN operation:

R(A1, A2, . . ., An) S(B1, B2, . . ., Bm)

R.Ai=S.Bj

Result is a relation Q with degree n + m attributes: Q(A1, A2, . . ., An, B1, B2, . . ., Bm), in that order.

The resulting relation state has one tuple for eachcombination of tuplesr from R and s from S, but only iftheysatisfy the join condition r[Ai]=s[Bj]

Hence, ifR has nR tuples, and S has nS tuples, then the joinresult will generally have less than nR * nS tuples.

Only related tuples (based on the join condition) will appearin the result


41/58

Slide 6- 41

Some properties of JOIN

The general case of JOIN operation is called aTheta-join: R S

theta

The join condition is called theta Theta can be any general boolean expression on

the attributes ofR and S; for example: R.Ai


42/58

Slide 6- 42

Binary Relational Operations: EQUIJOIN

EQUIJOIN Operation

The most common use of join involves joinconditions with equality comparisons only

Such a join, where the only comparison operatorused is =, is called an EQUIJOIN.

In the result of an EQUIJOIN we always have oneor more pairs of attributes (whose names need not

be identical) that have identical values in everytuple.

The JOIN seen in the previous example was anEQUIJOIN.


43/58

Slide 6- 43

Binary Relational Operations:NATURAL JOIN Operation

NATURAL JOIN Operation

Another variation of JOIN called NATURAL JOIN denoted by * was created to get rid of the second(superfluous) attribute in an EQUIJOIN condition.

because one of each pair of attributes with identical values issuperfluous

The standard definition of natural join requires that the twojoin attributes, or each pair of corresponding join attributes,have the same name in both relations

If this is not the case, a renaming operation is applied first.


44/58

Slide 6- 44

Binary Relational OperationsNATURAL JOIN (contd.)

Example: To apply a natural join on the DNUMBER attributes ofDEPARTMENT and DEPT_LOCATIONS, it is sufficient to write:

DEPT_LOCSn DEPARTMENT * DEPT_LOCATIONS

Only attribute with the same name is DNUMBER

An implicit join condition is created based on this attribute:DEPARTMENT.DNUMBER=DEPT_LOCATIONS.DNUMBER

Another example: Q n R(A,B,C,D) * S(C,D,E)

The implicit join condition includes each pairof attributes with thesame name, ANDed together: R.C=S.C AND R.D = S.D

Result keeps only one attribute of each such pair: Q(A,B,C,D,E)


45/58

Slide 6- 45

Example of NATURAL JOIN operation


46/58

Slide 6- 46

Complete Set ofRelational Operations

The set of operations including SELECT W,PROJECT T , UNION , DIFFERENCE ,RENAME V, and CARTESIAN PRODUCT X is

called a complete setbecause any otherrelational algebra expression can be expressedby a combination of these five operations.

For example:

R S = (R S ) ((R S) (S R))

R S = W (R X S)


47/58

Slide 6- 47

Binary Relational Operations: DIVISION

DIVISION Operation The division operation is applied to two relations

R(Z) z S(X), where X subset Z. Let Y = Z - X (and hence Z= X Y); that is, let Y be the set of attributes ofR that are

not attributes of S.

The result of DIVISION is a relation T(Y) that includes atuple t if tuples tR appear in R with tR [Y] = t, and with tR [X] = ts for every tuple ts in S.

For a tuple t to appear in the result T of the DIVISION, thevalues in t must appear in R in combination with everytuplein S.


48/58

Slide 6- 48

Example of DIVISION


49/58

Slide 6- 49

Recap ofRelational Algebra Operations


50/58

Slide 6- 50

Additional Relational Operations:Aggregate Functions and Grouping

A type of request that cannot be expressed in the basicrelational algebra is to specify mathematical aggregatefunctions on collections of values from the database.

Examples of such functions include retrieving the averageor total salary of all employees or the total number ofemployee tuples. These functions are used in simple statistical queries that

summarize information from the database tuples.

Common functions applied to collections of numericvalues include SUM, AVERAGE, MAXIMUM, and MINIMUM.

The COUNT function is used for counting tuples orvalues.


51/58

Slide 6- 51

Examples of applying aggregate functionsand grouping


52/58

Slide 6- 52

Illustrating aggregate functions andgrouping


53/58

Slide 6- 53

Additional Relational Operations (cont.)

The OUTER JOIN Operation

In NATURAL JOIN and EQUIJOIN, tuples without amatching(orrelated) tuple are eliminated from the joinresult

Tuples with null in the join attributes are also eliminated

This amounts to loss of information.

A set of operations, called OUTER joins, can be used whenwe want to keep all the tuples in R, or all those in S, or all

those in both relations in the result of the join,

regardless ofwhether or not they have matching tuples in the otherrelation.


54/58

Slide 6- 54


The left outer join operation keeps every tuple inthe first or left relation R in R S; if no matchingtuple is found in S, then the attributes of S in the

join result are filled or padded with null values.

A similar operation, right outer join, keeps everytuple in the second or right relation S in the resultofR S.

A third operation, full outer join, denoted bykeeps all tuples in both the left and the rightrelations when no matching tuples are found,padding them with null values as needed.


55/58

Slide 6- 55



56/58

Slide 6- 56


OUTER UNION Operations The outer union operation was developed to take

the union of tuples from two relations if therelations are not type compatible.

This operation will take the union of tuples in tworelations R(X, Y) and S(X, Z) that are partiallycompatible, meaning that only some of theirattributes, say X, are type compatible.

The attributes that are type compatible are

represented only once in the result, and thoseattributes that are not type compatible from eitherrelation are also kept in the result relation T(X, Y,Z).


57/58

Slide 6- 57


Example: An outer union can be applied to two relationswhose schemas are STUDENT(Name, SSN, Department,Advisor) and INSTRUCTOR(Name, SSN, Department,Rank). Tuples from the two relations are matched based on having the

same combination of values of the shared attributes Name,SSN, Department.

If a student is also an instructor, both Advisor and Rank willhave a value; otherwise, one of these two attributes will be null.

The result relation STUDENT_OR_INSTRUCTOR will have thefollowing attributes:

STUDENT_OR_INSTRUCTOR (Name, SSN, Department,Advisor, Rank)


58/58

Examples of Queries in RelationalAlgebra : Procedural Form

Q1: Retrieve the name and address of all employees who work for the

Research department.

RESEARCH_DEPTn W DNAME=Research (DEPARTMENT)

RESEARCH_EMPSn (RESEARCH_DEPT DNUMBER= DNOEMPLOYEEEMPLOYEE)

RESULTn T FNAME, LNAME, ADDRESS (RESEARCH_EMPS)

Q6: Retrieve the names of employees who have no dependents.

ALL_EMPSn T SSN(EMPLOYEE)

EMPS_WITH_DEPS(SSN)n T ESSN(DEPENDENT)

EMPS_WITHOUT_DEPS n (ALL_EMPS - EMPS_WITH_DEPS)

RESULTn T LNAME, FNAME (EMPS_WITHOUT_DEPS * EMPLOYEE)

rel algebra

Documents