concepts of relational algebra

8/7/2019 concepts of Relational Algebra

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Relational AlgebraRelational Algebra

Dashiell FryerDashiell Fryer


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W

hat is Relational Algebra?W

hat is Relational Algebra? Relational algebra is a procedural queryRelational algebra is a procedural query

language.language.

It consists of the select, project, union, setIt consists of the select, project, union, setdifference, Cartesian product, and renamedifference, Cartesian product, and renameoperations.operations.

Set intersection, division, natural join, andSet intersection, division, natural join, andassignment combine the fundamentalassignment combine the fundamentaloperations.operations.


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Select OperationSelect Operation Select operation returns a relation that satisfiesSelect operation returns a relation that satisfies

the given predicate from the original relation.the given predicate from the original relation.

Denoted by lowercase sigmaDenoted by lowercase sigma ..

Example:Example: branchbranch--name = Perryridgename = Perryridge(loan) from page 89(loan) from page 89returns a relation of the tuples from relationreturns a relation of the tuples from relation loanloanwhose branch is Perryridge.whose branch is Perryridge.

=,,,,,=,,,,, ,,v are legal in the predicate.,,v are legal in the predicate.


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ProjectProject Unary operationUnary operation

Denoted by uppercase piDenoted by uppercase pi

.. Returns a relation with only the specifiedReturns a relation with only the specified

attributes.attributes.

Example: page 90Example: page 90 loanloan--number, amountnumber, amount(loan) lists(loan) listsall of the loan numbers and the amounts.all of the loan numbers and the amounts.


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Composition of OperationsComposition of Operations Results of relational operations areResults of relational operations are

relations themselves.relations themselves.

Compositions of operations form aCompositions of operations form arelationalrelational--algebra expression.algebra expression.


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UnionUnion Denoted, as in set theory, byDenoted, as in set theory, by ..

B

inary operationB

inary operation Results in a relation with all of the tuplesResults in a relation with all of the tuples

that appear in either or both of thethat appear in either or both of theargument relations.argument relations.


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Union (Contd)Union (Contd)

Unions must be between compatibleUnions must be between compatiblerelationsrelations

Both relations must have the sameBoth relations must have the samenumber of attributes.number of attributes.

Domains of theDomains of the iith attribute of the firstth attribute of the first

and theand the iith attribute of the second mustth attribute of the second mustbe the same for allbe the same for all i.i.


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Set DifferenceSet Difference

Denoted byDenoted by --

B

inary operationB

inary operation RR S produces all tuples in R but not in SS produces all tuples in R but not in S

Relations must be compatible under theRelations must be compatible under thesame conditions as the union operation.same conditions as the union operation.


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CartesianCartesian--ProductProduct

Denoted by a cross (Denoted by a cross ())

B

inary operationB

inary operation Combines any two relationsCombines any two relations

RR S is the CartesianS is the Cartesian--Product of R and SProduct of R and S

Output has the attributes of both relationsOutput has the attributes of both relations


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CartesianCartesian--Product (Contd)Product (Contd)

Repeated attribute names are preceded byRepeated attribute names are preceded bythe relation they originated from.the relation they originated from.

Example: page 93 r= borrowerExample: page 93 r= borrower loanloan

(borrower.customer(borrower.customer--name,name,

borrower.loanborrower.loan--number, loan.loannumber, loan.loan--number,number,loan.branchloan.branch--name, loan.amount)name, loan.amount)


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RenameRename

Unary operationUnary operation

Denoted by lowercase Greek letter rhoDenoted by lowercase Greek letter rho

Useful for naming the unnamed relationsUseful for naming the unnamed relationsreturned from other operations.returned from other operations.


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Formal DefinitionFormal Definition

Relational Algebra acts on either a relationRelational Algebra acts on either a relationin the database or a constant relation.in the database or a constant relation.

Suppose ESuppose E11 and Eand E22 are relationalare relational--algebraalgebraexpressionsexpressions


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Formal Definition (Contd)Formal Definition (Contd)

All relationalAll relational--algebra expressions are:algebra expressions are:

EE11 EE22

EE11 EE22

EE11 EE22

PP(E

(E11

), P is the predicate), P is the predicateSS(E(E11), S is a list of attributes in E), S is a list of attributes in E11

xx(E(E11), x is the new name ofE), x is the new name ofE11


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Additional OperationsAdditional Operations

The fundamental operations can beThe fundamental operations can becomposed to make new, simplercomposed to make new, simpler

operations.operations.

SetSet--IntersectionIntersection

NaturalNatural--JoinJoin

DivisionDivision

AssignmentAssignment


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SetSet--IntersectionIntersection

Denoted by the symbolDenoted by the symbol ..

Results in a relation that contains only theResults in a relation that contains only thetuples that appear in both relations.tuples that appear in both relations.

RR S = RS = R (R(R S)S)

Since setSince set--intersection can be written inintersection can be written interms of setterms of set--difference, it is not adifference, it is not afundamental operation.fundamental operation.


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NaturalNatural--JoinJoin

Denoted by |x|.Denoted by |x|.

Binary operationBinary operation

Creates a CartesianCreates a Cartesian--product of theproduct of thearguments then performs selection toarguments then performs selection toforce equality on attributes that appear inforce equality on attributes that appear in

both relationsboth relations


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DivisionDivision

Denoted byDenoted by zz

Binary OperationBinary Operation

Used in queries that include the phraseUsed in queries that include the phrasefor all.for all.


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Division (Contd)Division (Contd)

Division is an operation on schema RDivision is an operation on schema R SS

A tuple t is in rA tuple t is in r zz s if and only if:s if and only if:

t is int is in RR SS(r) and(r) and

For every tuple tFor every tuple tss in s, there is a tuple tin s, there is a tuple trr ininr satisfying both of the following:r satisfying both of the following:

a. ta. trr[S] = t[S] = tss[R][R]

b. tb. trr[R[R S] = tS] = t


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AssignmentAssignment

Denoted by theDenoted by the nn symbolsymbol

Works like assignment operations inWorks like assignment operations inprogramming languagesprogramming languages

Assignment must always be made to aAssignment must always be made to atemporary relationtemporary relation

Does not result in display to the userDoes not result in display to the user


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SummarySummary

Relational Algebra defines a set ofRelational Algebra defines a set ofoperations that operate on relations andoperations that operate on relations and

result in new tables.result in new tables.

Operations can be composedOperations can be composed

Additional operations simplify the algebraAdditional operations simplify the algebra

but do not add any additional power.but do not add any additional power.


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The EndThe End

concepts of relational algebra

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