n-ary Relations and Their Applications

n-ary Relations and Their Applications

Rosen 8.2

n-ary relations

We can have relation between more than just 2 sets

A binary relation involves 2 sets and can be described by a set of pairsA ternary relation involves 3 sets and can be described by a set of triples…

An n-ary relation involves n sets and can be described by a set of n-tuples

Relations are used to represent computer databases

Also used to represent constraints!

n-ary relations n-ary Relations

ndomainsAAA

AAAaryn

AAA

n

n

n

isrelation theof degree Therelation theof theare ,,, sets The

product cartesian theofsubset a isrelation An

sets be ,,Let

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2,1


cbacbaNNNR

such that ),,( triplesof consisting on relation thebe Let

Note: N is the set of natural numbers {0,1,2,3,…}

An example

}),3,2,1(,),4,2,0(),3,2,0(,),3,1,0(),2,1,0{( R

R)3,4,2(

Note: R could be considered as an extensional representation of the ternary relation a<b<c, assuming domains are finite and really quite small

The relation has degree 3

The domains of the relation are the set of natural numbers


)0()(such that ),,,(tuples-4 of consisting on relation thebe Let

dcbadcbadcbaZNZNR

Note: N is the set of natural numbers {0,1,2,3,…} Z is the set of integers {…,-2,-1,0,1,2,…}

An example

Note: R could be considered as an extensional representation of the ternary relation a<b<c, assuming domains are finite and really quite small

The relation has degree 4RRR

)9,3,6,6()3,3,11,5(

)0,1,1,0(

Relational databases

Database is made up of records.Typical operations on a database are

• find records that satisfy a given criteria• delete records• add records• update records

Some everyday databases• student records• health records• tax information• telephone directories• banking records• …

Databases may be represented using the relational model

Relational databases The relational data model

Database made up of records, they are n-tuples, made up of fields

Student record might look as follows

(name,metricNo,faculty,gpa)

(Jones,200401986,Arts,4.9)(Lee,200408972,Science,3.6)(Kuhns,200501728,Humanities,5.0)(Moore,200308327,Science,5.5)

relations (in relDB) also called tables

gpa is an attribute


2.99Psychology786576Stevens3.90sMathematic678543Rao3.49sMathematic453876Goodfriend3.49ScienceComputer 102147Chou3.45Physics888323Adams3.88ScienceComputer 231455AckermannGPADeptmetricNoName

Example from the book

Attributes: name, metric No, Dept and GPA



primary key:An attribute/domain/column is a primary key whenthe value of this attribute uniquely defines tuplesi.e. no two tuples have the same value for that attribute

Name cannot be a primary key, neither can Dept or GPSmetricNo is a primary key



The current collection of n-tuples (records) in the relation (table)is called the extension of the relation

The permanent aspects of the relation (table) such as the attribute namesis called the intention of the relation



A composite key is a combination of attributesThat uniquely define tuples

Relational databases Operations on n-ary relations


CRtuplesnarynRS

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satisfy.must in elementsthat condition a andrelation an be Let

Selection



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3.90sMathematic678543Rao3.49sMathematic453876Goodfriend3.49ScienceComputer 102147Chou

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Projection

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4,1 projection Apply the P

2.99Stevens3.90Rao3.49Goodfriend3.49Chou3.45Adams3.88AckermannGPAName

Relational databases Operations on n-ary relationsJoin

Joins two tables/relations together, matching up on specific attributes

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RosenRosenGrammerGrammerFaberFaberCruzCruz

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Relational databases Operations on n-ary relationsJoin

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TimeRoomCourseDept

Relation

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),(2 SRJ

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n-ary Relations and Their Applications

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