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Relational Algebra & Relational Calculus

Session 5-6

Course : T0206 - Database System

Year : 2010/2011

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Outline Material Meaning of the term relational completeness.

How to form queries in relational algebra.

How to form queries in tuple relational calculus.

How to form queries in domain relational calculus.

Categories of relational DML.

Pearson Education 2009

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Introduction Relational algebra and relational calculus are

formal languages associated with the relationalmodel.

Informally, relational algebra is a (high-level)procedural language and relational calculus anon-procedural language.

However, formally both are equivalent to oneanother.

A language that produces a relation that can bederived using relational calculus is relationallycomplete.


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Relational Algebra Relational algebra operations work on one ormore relations to define another relation withoutchanging the original relations.

Both operands and results are relations, sooutput from one operation can become input toanother operation.

Allows expressions to be nested, just as inarithmetic. This property is called closure.


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Relational Algebra Five basic operations in relational algebra:Selection, Projection, Cartesian product, Union,and Set Difference.

These perform most of the data retrievaloperations needed.

Also have Join, Intersection, and Division

operations, which can be expressed in terms of 5basic operations.


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Relational Algebra Operations


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Relational AlgebraO

perations


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Selection (or Restriction) Wpredicate (R) Works on a single relation R and defines a relation that

contains only those tuples (rows) of R that satisfy thespecified condition (predicate).


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Example - Selection (or Restriction) List all staff with a salary greater than 10,000.

Wsalary > 10000 (Staff)


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Projection 4col1, . . . , coln(R) Works on a single relation R and defines a relation that

contains a vertical subset of R, extracting the values ofspecified attributes and eliminating duplicates.


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Example - Projection Produce a list of salaries for all staff, showing onlystaffNo, fName, lName, and salary details.

4staffNo, fName, lName, salary(Staff)


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Union R S Union of two relations R and S defines a relation that

contains all the tuples of R, or S, or both R and S, duplicatetuples being eliminated.

R and S must be union-compatible.

If R and S have Iand J tuples, respectively, union isobtained by concatenating them into one relationwith a maximum of (I+ J) tuples.


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Example - Union List all cities where there is either a branch office or a

property for rent.

4city(Branch) 4city(PropertyForRent)


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E

xample - Set Difference List all cities where there is a branch office but noproperties for rent.



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I

ntersection R S

Defines a relation consisting of the set of all tuples thatare in both R and S.

R and S must be union-compatible.

Expressed using basic operations:R S = R (R S)


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E

xample -I

ntersection List all cities where there is both a branch office andat least one property for rent.



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Cartesian product R X S Defines a relation that is the concatenation of every tuple of

relation R with every tuple of relation S.


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E

xample - Cartesian product List the names and comments of all clients who haveviewed a property for rent.

(4clientNo, fName, lName(Client)) X (4clientNo, propertyNo,

comment(Viewing))


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E

xample - Cartesian product andSelection Use selection operation to extract those tuples

where Client.clientNo = Viewing.clientNo.

WClient.clientNo = Viewing.clientNo

((4clientNo, fName, lName

(Client))' (4 clientNo, propertyNo, comment(Viewing)))

Cartesian product and Selection can be reduced to a single operation called a Join.


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JoinO

perations Join is a derivative of Cartesian product.

Equivalent to performing a Selection, using joinpredicate as selection formula, over Cartesianproduct of the two operand relations.

One of the most difficult operations to implementefficiently in an RDBMS and one reason why

RDBMSs have intrinsic performance problems.


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JoinO

perations Various forms of join operation Theta join

Equijoin (a particular type of Theta join)

Natural join

Outer join

Semijoin


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Theta join (U-join) R FS

Defines a relation that contains tuples satisfying thepredicate F from the Cartesian product of R and S.

The predicate F is of the form R.aiU S.b

iwhere U may be

one of the comparison operators (, u, =, {).


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Theta join (U-join) Can rewrite Theta join using basic Selection and

Cartesian product operations.

R FS = WF(R ' S)

Degree of a Theta join is sum of degrees of theoperand relations R and S. If predicate F containsonly equality (=), the term Equijoin is used.


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E

xample -E

quijoin List the names and comments of all clients who have

viewed a property for rent.(4clientNo, fName, lName(Client)) Client.clientNo = Viewing.clientNo (4clientNo,propertyNo, comment(Viewing))


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Natural join R S

An Equijoin of the two relations R and S over all commonattributes x. One occurrence of each common attribute iseliminated from the result.


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E

xample - Natural join List the names and comments of all clients who have

viewed a property for rent.(4clientNo, fName, lName(Client))

(4

clientNo, propertyNo, comment(Viewing))


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O

uter join To display rows in the result that do not havematching values in the join column, use Outerjoin.

R S (Left) outer join is join in which tuples from R that do

not have matching values in common columns of S arealso included in result relation.


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E

xample - LeftO

uter join Produce a status report on property viewings.

4propertyNo, street, city(PropertyForRent)

Viewing


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Semijoin R F S Defines a relation that contains the tuples of R that participate in

the join of R with S.

Can rewrite Semijoin using Projection and Join:

R F S = 4A(RF

S)


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E

xample - Semijoin List complete details of all staff who work at the branchin Glasgow.

Staff Staff.branchNo=Branch.branchNo(Wcity=Glasgow(Branch))


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Division R z S Defines a relation over the attributes C that

consists of set of tuples from R that matchcombination ofeverytuple in S.

Expressed using basic operations:T1 n 4C(R)

T2 n 4C((S X T1) R)

T n T1 T2


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E

xample - Division Identify all clients who have viewed all properties with

three rooms.

(4clientNo, propertyNo(Viewing)) z

(4propertyNo(Wrooms = 3 (PropertyForRent)))


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AggregateO

perations AL(R) Applies aggregate function list, AL, to R to define a relation

over the aggregate list.

AL contains one or more (,

) pairs . Main aggregate functions are: COUNT, SUM, AVG,

MIN, and MAX.


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E

xample AggregateO

perations How many properties cost more than 350 per

month to rent?

VR(myCount) COUNT propertyNo (rent > 350(PropertyForRent))


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GroupingO

peration GAAL(R) Groups tuples of R by grouping attributes, GA, and then

applies aggregate function list, AL, to define a new relation. AL contains one or more (,

) pairs. Resulting relation contains the grouping attributes, GA,

along with results of each of the aggregate functions.


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Example GroupingOperation Find the number of staff working in each branch

and the sum of their salaries.

VR(branchNo, myCount, mySum)

branchNo COUNT staffNo, SUM salary (Staff)


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Relational Calculus Relational calculus query specifies whatis to be

retrieved rather than how to retrieve it. No description of how to evaluate a query.

In first-order logic (or predicate calculus),predicateis a truth-valued function with arguments.

When we substitute values for the arguments,function yields an expression, called aproposition,which can be either true or false.


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Relational Calculus If predicate contains a variable (e.g. x is a memberof staff), there must be a range forx.

When we substitute some values of this range forx,

proposition may be true; for other values, it may befalse.

When applied to databases, relational calculus hasforms: tuple and domain.


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Tuple Relational Calculus Interested in finding tuples for which a predicate istrue. Based on use of tuple variables.

Tuple variable is a variable that ranges over anamed relation: i.e., variable whose only permittedvalues are tuples of the relation.

Specify range of a tuple variable Sas the Staffrelation as:

Staff(S)

To find set of all tuples S such that P(S) is true:{S | P(S)}


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Tuple Relational Calculus -E

xample To find details of all staff earning more than10,000:

{S | Staff(S) S.salary > 10000}

To find a particular attribute, such as salary,write:

{S.salary | Staff(S) S.salary > 10000}


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Tuple Relational Calculus Can use two quantifiers to tell how many instancesthe predicate applies to: Existential quantifier (there exists)

Universal quantifier (for all)

Tuple variables qualified by or are called boundvariables, otherwise called free variables.


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Tuple Relational Calculus Existential quantifier used in formulae that mustbe true for at least one instance, such as:

Staff(S) (B)(Branch(B)

(B.branchNo = S.branchNo) B.city = London)

Means There exists a Branch tuple with samebranchNo as the branchNo of the current Stafftuple, S, and is located in London.


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Tuple Relational Calculus Universal quantifier is used in statements aboutevery instance, such as:

(B) (B.city { Paris)

Means For all Branch tuples, the address is not inParis.

Can also use ~(B) (B.city = Paris) which meansThere are no branches with an address in Paris.


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Tuple Relational Calculus Formulae should be unambiguous and make sense. A (well-formed) formula is made out of atoms:

R(Si), where Si is a tuple variable and R is arelation

Si.a1 U Sj.a2 Si.a1 U c

Can recursively build up formulae from atoms: An atom is a formula

If F1 and F2 are formulae, so are their conjunction, F1 F2; disjunction, F1 F2; andnegation, ~F1

IfF is a formula with free variable X, then (X)(F)and (X)(F) are also formulae.


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Example - Tuple Relational Calculus List the names of all managers who earn more than

25,000.{S.fName, S.lName | Staff(S)

S.position = Manager S.salary > 25000}

List the staff who manage properties for rent inGlasgow.

{S | Staff(S) (P) (PropertyForRent(P) (P.staffNo = S.staffNo) P.city = Glasgow)}


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E

xample - Tuple Relational Calculus List the names of staff who currently do not manage

any properties.

{S.fName, S.lName | Staff(S) (~(P)(PropertyForRent(P)(S.staffNo = P.staffNo)))}

Or

{S.fName, S.lName | Staff(S) ((P) (~PropertyForRent(P)

~(S.staffNo = P.staffNo)))}


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E

xample - Tuple Relational Calculus List the names of clients who have viewed aproperty for rent in Glasgow.

{C.fName, C.lName | Client(C) ((V)(P)

(Viewing(V) PropertyForRent(P) (C.clientNo = V.clientNo)

(V.propertyNo=P.propertyNo)

P.city =Glasgow))}


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Tuple Relational Calculus Expressions can generate an infinite set. Forexample:{S | ~Staff(S)}

To avoid this, add restriction that all values inresult must be values in the domain of theexpression.


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E

xample - Domain Relational Calculus Find the names of all managers who earn morethan 25,000.

{fN, lN | (sN, posn, sex, DOB, sal, bN)

(Staff (sN, fN, lN, posn, sex, DO

B, sal, bN) posn = Manager sal > 25000)}


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Example - Domain Relational Calculus List the staff who manage properties for rent in

Glasgow.

{sN, fN, lN, posn, sex, DOB, sal, bN |

(sN1,cty)(Staff(sN,fN,lN,posn,sex,DO

B,sal,bN) PropertyForRent(pN, st, cty, pc, typ, rms,

rnt, oN, sN1, bN1)

(sN=sN1) cty=Glasgow)}


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Example - Domain Relational Calculus List the names of staff who currently do not

manage any properties for rent.

{fN, lN | (sN)

(Staff(sN,fN,lN,posn,sex,DO

B,sal,bN) (~(sN1) (PropertyForRent(pN, st, cty, pc, typ,

rms, rnt, oN, sN1, bN1) (sN=sN1))))}


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Example - Domain Relational Calculus List the names of clients who have viewed a

property for rent in Glasgow.

{fN, lN | (cN, cN1, pN, pN1, cty)

(Client(cN, fN, lN,tel, pT, mR, eM) Viewing(cN1, pN1, dt, cmt)

PropertyForRent(pN, st, cty, pc, typ,

rms, rnt,oN, sN, bN)

(cN = cN1) (pN = pN1) cty = Glasgow)}


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Domain Relational Calculus When restricted to safe expressions, domain

relational calculus is equivalent to tuple relationalcalculus restricted to safe expressions, which isequivalent to relational algebra.

Means every relational algebra expression has anequivalent relational calculus expression, and viceversa.


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Other Languages Transform-oriented languages are non-procedural

languages that use relations to transform input datainto required outputs (e.g. SQL).

Graphical languages provide user with picture of thestructure of the relation. User fills in example of whatis wanted and system returns required data in thatformat (e.g. QBE).


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Other Languages 4GLs can create complete customized application

using limited set of commands in a user-friendly,often menu-driven environment.

Some systems accept a form ofnatural language,sometimes called a 5GL, although this developmentis still at an early stage.

P Ed ti 2009

t02060010220104031t0206 - pert 5-6

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