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© by Kenneth H. Rosen, Discrete Mathematics & its Applications, Sixth Edition, Mc Graw-Hill, 2007

Chapter 8: Relations

Relations(8.1)

n-any Relations & their Applications (8.2)

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Relations (8.1)

Introduction

– Relationship beteen a pro!ra" and its

#ariables

– Inte!ers that are con!ruent "odulo $

– %airs o cities lin$ed by airline li!hts in a

netor$

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Relations (8.1) (cont.)

Relations & their properties

– 'einition 1

et A and be sets. A binary relation ro" A to is a subset o A * .

In other ords+ a binary relation ro" A to isa set R o ordered pairs here the irstele"ent o each ordered pair co"es ro" Aand the second ele"ent co"es ro" .

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– ,otation:

aRb ⇔ (a+ b) ∈ R

aRb ⇔ (a+ b) ∉ R

R a b

0

1

2

X X

X

X

0

1

2

a

b

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– a"ple: A / set o all cities

/ set o the 0 states in the 3A

'eine the relation R by speciyin! that (a+ b)

belon!s to R i city a is in state b.

R.inare

) New Jersey Bank, Red

)California ,(Cupertino

) Michigan , Aror Ann( ) Maine , Bangor (

)Colorado , Boulder (

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4unctions as relations

– 5he !raph o a unction is the set o orderedpairs (a+ b) such that b / (a)

– 5he !raph o is a subset o A * ⇒ it is arelation ro" A to

– Con#ersely+ i R is a relation ro" A to suchthat e#ery ele"ent in A is the irst ele"ent oeactly one ordered pair o R+ then a unctioncan be deined ith R as its !raph

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Relations on a set

– 'einition 2

A relation on the set A is a relation ro" A to A.

– a"ple: A / set 61+ 2+ 7+ 9. hich ordered pairs arein the relation R / 6(a+ b) ; a di#ides b9

Solution: 3ince (a+ b) is in R i and only i a and b arepositi#e inte!ers not eceedin! such that a di#ides b

R / 6(1+1)+ (1+2)+ (1.7)+ (1.)+ (2+2)+ (2+)+ (7+7)+ (+)9

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%roperties o Relations

– 'einition 7

A relation R on a set A is called relei#e i

(a+ a) ∈ R or e#ery ele"ent a ∈ A.

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– a"ple (a): Consider the olloin! relationson 61+ 2+ 7+ 9

R1 / 6(1+1)+ (1+2)+ (2+1)+ (2+2)+ (7+)+ (+1)+ (+)9

R2 / 6(1+1)+ (1+2)+ (2+1)9R7 / 6(1+1)+ (1+2)+ (1+)+ (2+1)+ (2+2)+ (7+7)+ (7+)+ (+1)+(+)9R8 / 6(2+1)+ (7+1)+ (7+2)+ (+1)+ (+2)+ (+7)9R0 / 6(1+1)+ (1+2)+ (1+7)+ (1+)+ (2+2)+ (2+7)+ (2+)+ (7+7)+

(7+)+ (+)9R

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Solution:

R7 and R0: relei#e ⇐ both contain all pairs

o the or" (a+ a): (1+1)+ (2+2)+ (7+7) & (+).

R1+ R2+ R8 and R

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– 'einition :

A relation R on a set A is called sy""etric i

(b+ a) ∈ R hene#er (a+ b) ∈ R+ or all a+b ∈ A.

A relation R on a set A such that (a+ b) ∈ R

and (b+ a) ∈ R only i a / b+ or all a+ b ∈ A+ is

called antisy""etric.

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– a"ple: hich o the relations ro" ea"ple (a) aresy""etric and hich are antisy""etric=

Solution:

R2 & R7: sy""etric ⇐ each case (b+ a) belon!s to the relation

hene#er (a+ b) does.

4or R2: only thin! to chec$ that both (1+2) & (2+1) belon! to the

relation

4or R7: it is necessary to chec$ that both (1+2) & (2+1) belon! to

the relation.

,one o the other relations is sy""etric: ind a pair (a+ b) so

that it is in the relation but (b+ a) is not.

R1 = {(1,1), (1,2), (2,1), (2,2), (3,4), (4,1), (4,4)}

R2 = {(1,1), (1,2), (2,1)}

R3 = {(1,1), (1,2), (1,4), (2,1), (2,2), (3,3), (4,1), (4,4)}

R4 = {(2,1), (3,1), (3,2), (4,1), (4,2), (4,3)}

R5 = {(1,1), (1,2), (1,3), (1,4), (2,2), (2,3), (2,4), (3,3), (3,4), (4,4)}R6 = {(3,4)}

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Solution (cont.):

R8+ R0 and R

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– 'einition 0:

A relation R on a set A is called transiti#e ihene#er (a+ b) ∈ R and (b+c) ∈ R+ then

(a+ c) ∈ R+ or all a+ b+ c ∈ R.

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– a"ple: hich o the relations in ea"ple (a) aretransiti#e=

R + R0 & R

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Co"binin! relations

– a"ple:

et A / 61+ 2+ 79 and / 61+ 2+ 7+ + 9. 5he relations R1 / 6(1+1)+ (2+2)+ (7+7)9 and

R2 / 6(1+1)+ (1+2)+ (1+7)+ (1+)9 can be co"bined to

obtain:

R1 ∪ R2 / 6(1+1)+ (1+2)+ (1+7)+ (1+)+ (2+2)+ (7+7)9

R1∩ R2 / 6(1+1)9

R1 – R2 / 6(2+2)+ (7+7)9

R2 – R1 / 6(1+2)+ (1+7)+ (1+)9

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– 'einition

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– a"ple: hat is the co"posite o the relations Rand 3 here R is the relation ro" 61+2+79 to 61+2+7+9ith R / 6(1+1)+ (1+)+ (2+7)+ (7+1)+ (7+)9 and 3 is therelation ro" 61+2+7+9 to 6+1+29 ith 3 / 6(1+)+ (2+)+(7+1)+ (7+2)+ (+1)9=

Solution: 3 ° R is constructed usin! all ordered pairsin R and ordered pairs in 3+ here the secondele"ent o the ordered in R a!rees ith the irst

ele"ent o the ordered pair in 3.4or ea"ple+ the ordered pairs (2+7) in R and (7+1) in3 produce the ordered pair (2+1) in 3 ° R. Co"putin!all the ordered pairs in the co"posite+ e ind

3 ° R / ((1+)+ (1+1)+ (2+1)+ (2+2)+ (7+)+ (7+1)9

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,-ary Relations & their Applications (8.2)

Relationship a"on! ele"ents o "ore

than 2 sets oten arise: n-ary relations

Airline+ li!ht nu"ber+ startin! point+

destination+ departure ti"e+ arri#al ti"e

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,-ary Relations & their Applications (8.2) (cont.)

,-ary relations

– 'einition 1:

et A1+ A2+ >+ An be sets. An n-ary relation on thesesets is a subset o A1 * A2 *>* An here Ai are the

do"ains o the relation+ and n is called its de!ree.

– a"ple: et R be the relation on , * , * , consistin!o triples (a+ b+ c) here a+ b+ and c are inte!ers ith

a?b?c. 5hen (1+2+7) ∈ R+ but

(2++7) ∉ R. 5he de!ree o this relation is 7. Its

do"ains are e@ual to the set o inte!ers.

21

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'atabases & Relations

– Relational database "odel has been de#elopedor inor"ation processin!

– A database consists o records+ hich are n-tuples "ade up o ields

– 5he ields contains inor"ation such as: ,a"e

3tudent B

aDor

Erade point a#era!e o the student

22

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– 5he relational database "odel represents a

database o records or n-ary relation

– 5he relation is R(3tudent-,a"e+ Id-nu"ber+

aDor+ E%A)

2

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– a"ple o records

(3"ith+ 721+ athe"atics+ 7.F)

(3te#ens+ 112+ Co"puter 3cience+ .)

(Rao+

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Students

Names

ID # Ma!" $%

3"ith

3te#ens

Rao Ada"s

ee

721

112

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Hperations on n-ary relations

– 5here are #arieties o operations that are appliedon n-ary relations in order to create ne relationsthat anser e#entual @ueries o a database

– 'einition 2:

et R be an n-ary relation and C a condition thatele"ents in R "ay satisy. 5hen the selection

operator sC "aps n-ary relation R to the n-aryrelation o all n-tuples ro" R that satisy thecondition C.

2/

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– a"ple:

i sC / aDor / co"puter scienceJ ∧ E%A K

7.0J then the result o this selection consists

o the 2 our-tuples:

(3te#ens+ 112+ Co"puter 3cience+ .)

(ee+ 17+ Co"puter 3cience+ 7.G)

27

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– 'einition 7:

5he proDection "aps the n-tuple

(a1+ a2+ >+ an) to the "-tuplehere " ≤ n.

In other ords+ the proDectiondeletes n – " o the co"ponents o n-tuple+

lea#in! the i1th+ i2th+ >+ and i"th co"ponents.

!i ,...,"i ,#i $

)a ,...,a ,a( !i "i #i

!i ,...,"i ,#i $

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– 'einition :

et R be a relation o de!ree " and 3 a relation

o de!ree n. 5he Doin Lp(R+3)+ here p≤ " andp ≤ n+ is a relation o de!ree

" M n – p that consists o all (" M n – p)-tuples

(a1+ a2+ >+ a"-p+ c1+ c2+ >+ cp+ b1+ b2+ >+ bn-p)+

here the "-tuple (a1+ a2+ >+ a"-p+ c1+ c2+ >+ cp)belon!s to R and the n-tuple (c1+ c2+ >+ cp+ b1+

b2+ >+ bn-p) belon!s to 3.

0

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– a"ple: hat relation results hen theoperator L2 is used to co"bine the relationdisplayed in tables C and '=

1%roessor 'pt Course B

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1%roessor 'pt Course B

CruN

CruN

4arber

4arber Era""er

Era""er

Rosen

Rosen

Ooolo!y

Ooolo!y

%sycholo!y

%sycholo!y%hysics

%hysics

Co"puter 3cience

athe"atics

770

12

01

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Solution: 5he Doin L2 produces the relationshon in 5able

%roessor 'pt Course B Roo" 5i"e

CruN

CruN

4arber

4arber

Era""er Rosen

Rosen

Ooolo!y

Ooolo!y

%sycholo!y

%sycholo!y

%hysicsCo"puter 3cience

athe"atics

770

12

01

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– a"ple: e ill illustrate ho 3P (3tructured Pueryan!ua!e) is used to epress @ueries by shoin! ho3P can be e"ployed to "a$e a @uery about airlineli!hts usin! 5able 4. 5he 3P state"ents

SELECT departure_time

FROM Flights

WHERE destination = ‘Detroit

are used to ind the proDection %0 (on the departureQti"e

attribute) o the selection o 0-tuples in the li!htsdatabase that satisy the condition: destination /'etroitS. 5he output ould be a list containin! the ti"eso li!hts that ha#e 'etroit as their destination+ na"ely+8:1+ 8:G+ and F:.

(

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Airline 4li!ht B Eate 'estination 'eparture ti"e

,adir Ac"e

Ac"e

Ac"e

,adir Ac"e

,adir

122221

122

727

1FF222

722

722

77

7

1722

7

'etroit'en#er

Anchora!e

Tonolulu

'etroit'en#er

'etroit

8:18:1G

8:22

8:7

8JGF:1

F:

able * *li6hts

210-ch8-part1

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