mc0063(a) unit1 final

DiscreteMathematics Unit1

SikkimManipalUniversity PageNo:1

Unit1 Sets,RelationsandFunctionsStructure

1.1 Introduction

Objectives

1.2 Sets

1.3 Relations

1.4 Functions

1.5 Intervals

1.6 Functionsofrealvariables

1.7 Differentfunctions

SelfAssessmentQuestions

1.8 Summary

1.9 TerminalQuestions

1.10Answers

1.1Introduction

Theconceptsofset,relationandfunctionareoffundamentalimportancein

modern mathematics. The idea of a set has been intuitively used in

mathematics since the time of ancient Greeks now set theory and its

associated branches such as Group theory, Ring theory etc., have far

reachingapplications.

The systematic development of set theory is attributed to the German

mathematicianGeorgeCantor(18451918).

Someelementarydefinitionsofsettheoryhavebeenstudiedbystudentsin

thehighschoolstandard.Inthischapterwebrieflygivesomepreliminaries

ofsettheoryanddiscusstherelationsandfunctions.



Objectives

Attheendofthisunitthestudentshouldbeableto:

Performdifferentoperationsonsets

UseVenndiagrams

Describedifferenttypesofrelationsandfunctions

1.2Sets

Thenotionofasetiscommon,intuitivelyasetisawelldefinedcollectionof

objects.Theobjectscomprisingthesetarecalled itsmemberorelements.

The sets areusually denoted by the capital letter A,B,X,Y etc., and its

elementsbysmalllettersa,b,x,yThestatementxisanelementofA

isdenotedbyx A andisreadasxbelongstoA.Ifxisnotanelementof

the set A then it is denoted by x A (read as x does not belong to A).

Whenever possible a set is written by enclosing its elements by brace

brackets{}.Forexample, A={a,e,i,o,u}.Theotherwayofspecifyingthe

setisstatingthecharacteristicpropertysatisfiedbyitselements.Theabove

exampleof thesetcanalsobewrittenasthesetofvowels in theEnglish

alphabetandiswrittenas,

A={x/x isavowelintheEnglishalphabet}.

Ifthenumberofelementsinasetisfinitethenitsissaidtobeafiniteset,

otherwiseitissaidtobeaninfiniteset.Ifasetcontainsonlyoneelementit

iscalledasingletonset.Ifasetcontainsnoelementsitiscalledanullsetor

emptyset,denotedby f.

Forexample,

B={1,3,5,7,9}isafiniteset,

N={x:x isanaturalnumber}={1,2,3,4,..}isaninfiniteset,

C={2} isasingletonsetand

D={x:x2=9andxiseven}isanemptyset.

Asetconsistingofatleastoneelementiscalledanonemptyset.



1.2.1Definition

IfeveryelementofasetAisalsoanelementofasetB thenAissaidtobe

a subset of B and it is denoted by BA or AB . Clearly A F ,

AA .

Forexample,letN,Z,Q,R respectivelydenotethesetofnaturalnumbers

the set of integers the set of rational numbers the set of real numbers.

Then

RQZN .

AsetA issaid tobeapropersubsetofB if thereexistsanelementofB

whichisnotanelementofA.ThatAisapropersubsetofB if BA and

BA .

Forexample,ifA={1,3,5},B={1,3,5,7} thenAisapropersubsetofB

Twosets AandBaresaidtobeequalifandonlyif BA and AB .

1.2.2Familyofsets

If theelementsofasetAarethemselvessetsthenA iscalledafamilyof

setsoraclassofsets.ThesetofallsubsetsofasetA iscalledthepower

setofAanditisdenotedbyP(A).

ForexampleifA={1,3,5}then.

P(A)={{ f},{1),{3),{5},{1,3},{3,5},{5,1}{1,3,5}}

Notethatthereare23=8 elementsinP(A).IfasetA hasn elementsthen

itspowersetP(A) has2nelements.

1.2.3Cardinalityofaset

IfAisafiniteset,thenthecardinalityofA isthetotalnumberofelements

thatcomprisethesetandisdenotedbyn(A).



The cardinal number or cardinality of each ofthe sets { } { } { } c,b,a,b,a,a, F isdenotedby0,1,2,3,respectively.

1.2.4Universalset

Inanydiscussion ifall thesetsaresubsetsofa fixedset, then thisset is

calledtheuniversalsetandisdenotedbyU..

For example, in the study of theory of numbers the set Z of integers is

consideredastheuniversalset.

1.2.5Unionofsets

Theunionof twosets AandB denotedby BA is the set ofelements

whichbelongtoAorBorboth.

Thatis, { } BxorAx:xBA = .Properties

1. AAA = 2. ABBA =

3. ( ) ( ) CBACBA = 4. BAA and BAB 1.2.6Intersectionofsets

The intersection of two sets A and B denoted by BA is the set of

elementswhichbelongtobothAandB.

Thatis, { } BxandAx:xBA = .Properties

1. AAA =

2. ABBA =

3. ( ) ( ) CBACBA = 4. ( ) ( ) ( ) CABACBA = 5. ( ) ( ) ( ) CABACBA =

If F = BA thenAandBaresaidtobedisjointsets.



1.2.7 Differenceofsets

ThedifferenceoftwosetsAandB,denotedbyABisthesetofelements

of A whicharenottheelementsofB.Thatis,

{ } Bx,Ax:xBA = - Clearly,

1. ABA -

2. ABBA - -

3. AB,BA,BA - - aremutuallydisjointsets.

1.2.8Complementofaset

ThecomplementofasetAwithrespecttotheuniversalsetUisdefinedas

UAandisdenotedbyA or cA .Thatis,

{ } Ax,Ux:xA = Clearly,

1. ( ) AA = 2. U = F 3. F = U1.2.9 DeMorganslaws

ForanythreesetsA,B,C

1. ( ) ( ) ( ) CABACBA - - = - 2. ( ) ( ) ( ) CABACBA - - = - 3. ( ) BABA = 4. ( ) BABA = 1.2.10Cartesianproductoftwosets

LetAandBbetwosets.ThentheCartesianproductofAandBisdefined

as the set of all ordered pairs. (x, y) Where ByandAx and is

denotedbyAx B.



Thus,

( ) { } ByandAx:y,xBA = Twoorderedpairs(a,b)and(c,d)areequalifandonlyifa=cand b=d.

IfA containsmelementsandBcontainsnelementsthenAxBcontains

mn orderedpairs.

1.2.11Example IfA={2,3,4},B={4,5,6}andC={6,7}

Evaluatethefollowing

(a) ( ) ( ) CBBA - (b) ( ) BCA - (c) ( ) BBA - (d) ( ) ( ) CBBA - - (e) ( ) ( ) CBBA - Solution:

(a) { } { } { } 46,5,44,3,2BA = = { } { } { } 5,47,66,5,4CB = - = -

Therefore ( ) ( ) { } { } ( ) ( ) { } 5,4,4,45,44CBBA = = - (b) { } { } { } { } 76,5,47,6BC4,3,2A = - = - =

Therefore ( ) { } { } ( ) ( ) ( ) { } 7,47,37,274,3,2BCA = = - (c) { } { } { } 3,26,5,44,3,2BA = - -

Therefore ( ) { } { } 6,5,43,2BBA = - ( ) ( ) ( ) ( ) ( ) ( ) { } 6,3,5,3,4,3,6,2,5,2,4,2 =

(d)Wehave ( ) ( ) { } { } 5,43,2CBBA = - - ( ) ( ) ( ) ( ) { } 5,34,35,24,2 =

(e) { } { } 6,5,44,3,2BA = ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) { } 6,4,5,4,4,4,6,3,5,3,4,3,6,2,5,2,4,2 =

{ } { } 7,66,5,4CB = ( ) ( ) ( ) ( ) ( ) ( ) { } 7,6,6,6,7,5,6,5,7,4,6,4 =



Therefore ( ) ( ) ( ) ( ) { } CBb,a:BAb,aCBBA = - ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) { } 5,4,4,4,6,3,5,3,4,3,6,2,5,2,4,2 =

1.2.12Example: If { } { } { } { } e,c,bC,e,dB,d,c,aA,e,d,c,b,aU = = = = Evaluatethefollowing

(a) ( ) CBA - (b) ( ) ( ) CBBA (c) ( ) ( ) CBBA - - (d) ( ) ACB (e) ( ) CAB - Solution:

(a) { } { } { } e,bd,c,ae,d,c,b,aAUA = - = - = { } { } { } de,c,be,dCB = - = -

( ) { } { } ( )( ) { } d,ed,bde,bCBATherefore = = - (b) ( ) ( ) { } { } { } be,d,c,ae,d,c,b,aBAUBA = - = - =

{ } { } { } ee,c,be,dCB = = ( ) ( ) { } { } ( ) { } e,bebCBBATherefore = =

(c) { } { } { } c,ae,dd,c,aBA = - = - { } { } { } de,c,be,dCB = - = -

( ) ( ) { } { } ( )( ) { } d,cd,adc,aCBBATherefore = = - - (d) { } { } { } e,d,c,be,c,be,dCB = =

( ) ( ) { } aCBUCBTherefore = - = ( ) { } { } d,c,aaACBTherefore =

( ) ( ) ( ) { } d,a,c,a,a,a =



(e) { } { } { } ed,c,ae,dAB = - = - { } { } d,ae,c,bUCUC = - = - =

Therefore ( ) { } { } ( ) ( ) { } d,e,a,ed,aeCAB = = - 1.2.13Example

If

= + - = 06x5x:xA 2

{ } 4,3,0B = { } 4xandNx:xC < =

Evaluatethefollowing:

(a) ( ) CBA (b) ( ) ( ) CBBA - (c) ( ) ( ) BCBA - - Solution:Now ( )( ) { } { } 3,203x2x:xA = = - - =

{ } 4,3,0B = { } { } 3,2,14xandNx:xC = < =

(a) { } { } { } { } 33,2,14,3,0CB3,2A = = = Therefore ( ) { } { } ( )( ) { } 3,33,233,2CBA = =

(b) { } { } { } 4,3,2,04,3,03,2BA = = { } { } { } 4,03,2,14,3,0CB = - = -

Therefore ( ) ( ) { } { } 4,04,3,2,0CBBA = - ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) { } 4,4,0,4,4,3,0,3,4,2,0,2,4,0,0,0 =

(c) { } { } { } 24,3,03,2BA = - = - { } { } { } 2,14,3,03,2,1BC = - = -

Therefore ( ) ( ) { } { } ( ) ( ) { } 2,2,1,22,12BCBA = = - -



1.3Relations

LetA andBbetwononemptysets.ThenarelationR from A toB isa

subsetofAxBcontainingtheorderedpairs ( ) BAb,a suchthatsomerelationexistsbetweenaandb.

InotherwordsarelationRismerelyasubsetofAxB.

If ( ) Rb,a thenwesaythataisR relatedtobandiswrittenasaRb.IfB=AthenwesaythatRisarelationinA.

Forexample,let A={2,3,5}B=(4,6,9}then

AxB={(2,4),(2,6),(2,9),(3,4),(3,6),(3,9),(5,4),(5,6),(5,9)}

DefinearelationRby

R= A:BA)b,a{( dividesb}

Then R= ( ) ( ) ( ) ( ) { } 9,3,6,3,6,2,4,2IfanotherrelationSisdefinedby

S= a:BA)b,a({ andbarerelativelyprime}

Then S={(2,9)(3,4)(5,4),(5,6),(5,9)}

1.3.1Domainandrangeofarelation

Let R be a relation from A to B. The domain of R is the set of all first

coordinates of the ordered pairs of R and the range of R is the set of

secondcoordinatesofthepairsofR.

Thatis, Domainof ( ) { } Rb,a:AaR = Rangeof ( ) { } Rb,a:BbR =

Obviouslythedomainof Risasubsetof AandtherangeofR isasubset

of B.

Intheaboveexample,domainofR={2,3}rangeofR={4,6,9}



1.3.2Inverseofarelation

Let R be a relation from a set A to a set B. The inverse relation of R

denotedbyR1 istherelationfromBtoAdefinedby

( ) ( ) { } Rb,a:a,bR 1 = -

InotherwordsR1 canbeobtainedbyreversingtheorderedpairsofR.

ClearlyR1isasubsetof AB .

Forexample,letA={2,3,5},B={1,4}

If ( ) { } ba:BAb,aR > = then ( ) ( ) ( ) ( ) { } 4,5,1,5,1,3,1,2R = Then ( ) ( )( ) ( ) { } 5,45,13,12,1R 1 = -

Nowweprovidethetypesofrelations

1.3.3Reflexiverelation

ArelationRinasetAissaidtobereflexiveifforevery ( ) Aa,a,Aa .ThusR isreflexiveifwehaveaRaforevery Aa .

Examples:

1. LetLbethesetoflinesintheplane.ConsiderarelationRdefinedby

R={(x,y):xisparalleledtoy}

Sinceeverylineisparalleltoitself,itfollowsthat ( ) Rx,x foreveryRx .HenceRisareflexiverelation.

2. LetN bethesetnaturalnumbers.RelationsxRydefinedby

i) xdividesyisreflexivesinceeverynaturalnumberdividesitself.

ii) x=yisreflexivesinceeverynumberisequaltoitself.

iii) xyisdivisibleby5isreflexivesincexx=0isdivisibleby5for

every Nx .

iv) x



1.3.4Symmetricrelation

ArelationRinasetAissaidtobesymmetricif ( ) Rb,a implies ( ) Ra,b ThatisRissaidtobesymmetricifaRbimpliesbRa

Examples

1. LetAbethesetoftrianglesinaplane.TherelationinAdefinedbya is

similarto b where A, b a issymmetricsinceifatriangle a is

similarto b then b isalsosimilarto a.

SimilarlyifLissetofstraightlinesinaplanethenarelationinLdefined

byxisparalleltoyandxisperpendiculartoyaresymmetric.

2. DefinerelationR in N by

i) x=yisasymmetricrelationsince x=y impliesy=x.

ii) xyisdivisibleby5isasymmetricrelationsinceifxyis

divisibleby5thenyxisalsodivisibleby5.

iii) x



Then R isnot an anti symmetric relation inA since ( ) R2,1 and ( ) R1,2 but 21

1.3.6Transitiverelation

ArelationRinasetAissaidtobetransitiveif ( ) ( ) Rc,b,Rb,a implies ( ) Rc,a ThatisRissaidtobetransitiveifaRb andbRcimpliesaRc.

Examples

1. IfListhesetofallstraightlinesinaplanethenarelationinLdefinedby

aisparalleltobistransitivesinceifa,b,carethreestraightlinesin

L thena isparallel tobandb isparallel toc impliesa isparallel toc.

Howeverarelationdefinedbyaisperpendiculartobisnottransitive

since aisperpendiculartobandbisperpendiculartocdoesnotimply

aisperpendiculartocinfacta isparalleltoc.

2. TherelationsinNdefinedby

i) x=yistransitivesince x=yandy=zimpliesx=z

ii) xyisdivisibleby5istransitivesinceifxyisdivisibleby5

andyzisdivisibleby5impliesxy+yz=xzisdivisibleby

5.

iii) x



Examples

1. IfA isthesetoftrianglesinaplanethentherelationR definedbyais

similartobisanequivalencerelation.

For,if a,b,careanythreetriangles,then(i)aissimilartoitself(ii)ifa

issimilartobthenbissimilartoa(iii)ifaissimilartobandbissimilar

to cthenaissimilartoc.

2. In the setLofall straight lines inaplane the relationdefinedby a is

parallel to b is an equivalence relation. For if a, b, c are any three

straightlinesthen(i)a isparalleltoitself(ii)ifa isparalleltob thenb

is parallel to a (iii) if a is parallel to b and b is parallel to c then a is

paralleltoc.

However the relation a is perpendicular to b is not an equivalence

relation.Sinceitissymmetricbutnotreflexiveandtransitive.

3. LetNbethesetofnaturalnumbers.Therelationsdefinedby

i) x=yisanequivalencerelation.

ii) xyisdivisibleby5isanequivalencerelation.

iii) x



functionf,theny iscalledtheimageofxunderfandisdenotedbyy=f(x).

Alsoxiscalledthepreimageofyunderf

TherangeoffisthesetofthoseelementsofBwhichappearastheimage

of at least one element of A and is denoted by f(A). Thus

( ) ( ) { } Ax:BxfAf = .Clearly f(A)isasubsetofB.1.4.2Example:LetA={1,2,3,4}andZbethesetofintegers.Define

ZA:f byf(x)=2x+3.ShowthatfisafunctionfromAtoB.Alsofind

therangeoff.

Solution:

Now ( ) ( ) ( ) ( ) 114f,93f,72f,51f = = = = Therefore ( ) ( ) ( ) ( ) { } 11,4,9,3,7,2,5,1f = SinceeveryelementofA isassociatedwithauniqueelementofB, f isa

function.

Rangeof f={5,7,9,11}

1.4.3 Example: Let N be the set of natural numbers. If NN:f is

definedbyf(x)=2x1showthatfisafunctionandfindtherangeoff.

Solution:

Now ( ) ( ) ( ) ,.....53f,32f,11f = = = Therefore ( ) ( ) ( ) ( ) { } ,.....7,4,5,3,3,2,1,1f = Clearly f isafunction.

Rangeoff={1,3,5,7,...}

1.4.4Example: LetRbe the set of realnumbers.Define RR:f by

( ) 2xxf = forevery Rx .Showthatfisafunctionandfindtherangeoff.



Solution:

Here fassociateseveryrealnumbertoitssquare,whichiscertainlyareal

number.Hencef isafunction.RangeofR is thesetofallnonnegative

realnumbers.

1.4.5OneOnefunction

A function BA:f is said to be one one or injection if for all

( ) ( ) 2121 xfxf,A,x,x = implies 21 xx = . The contrapositive of thisimplicationisthatforall 2121 xx,A,x,x implies ( ) ( ) 21 xfxf .

Thusafunction BA:f issaidtobeoneoneifdifferentelementsof

AhavedifferentimagesinB.

1.4.6Example:LetRbethesetofrealnumbers.Define RR:f by

i) ( ) 3x2xf + = (ii) ( ) 3xxf = forevery Rx provethatfisoneone.Solution:

i) Let ( ) ( ) 21 xfxf = forsome Rx,x 21 3x23x2 21 + = +

21 xx =

Thus for every ( ) ( ) 2121 xfxf,Rx,x = implies 21 xx = .Therefore f isoneone.

ii) Let ( ) ( ) 21 xfxf = forsome Rx,x 21 32

31 xx =

21 xx = Thereforef isoneone.

1.4.7Example: Iff:R R isdefinedby ( ) 2xxf = forevery Rx ,showthatfisnotoneone.



Solution:

Let ( ) ( ) 21222121 xxxxxfxf = = = Hence f isnotoneone.

Forexample, ( ) 42f = - and ( ) 42f = .The imagesof2and2arenotdifferent.Hence f isnotoneone.

1.4.8OntoFunction

A function BA:f is said tobe onto or surjection if for every By

there exist at least one element Ax such that f(x) = y. i.e., every

elementofthecodomainBappearsastheimageofatleastoneelement

ofthedomain A.

If fisontothen f(A)=B

1.4.9Example: Define RR:f by

(i) f(x)=2x+3 (ii) ( ) 3xxf = forever Rx Showthatfisonto

Solution

i) Let Ry .Thentofind Rx suchthatf(x)=y i.e.,2x+3=y

Solvingforxweget,2

3yx - =

Since R2

3yx,Ry - =

Henceforevery Ry exists R2

3yx

- = such

that y2

3yf =

- .Thereforefisonto.



ii) Let Ry . We shall show that there exists Rx such that

f (x)=y.That is yx3 = .Hence 31

yx = . If Ry , then Ry31

.

Thus for every Ry there exists Ry31

such that

yyyf

3

31

31

=

=

.

Thereforefisonto.

1.4.10Example: If RR:f isdefinedby ( ) 2xxf = forevery Rx thenprovethatfisnotonto.

Solution:

Sinceanegativenumberisnotthesquareofanyrealnumber,thenegative

numbersdonotappearastheimageofanyelementofR.

For example, R9 - but there does not exist any Rx such that

( ) .9xxf 2 - = = Hencefisnotonto.1.4.11OnetooneFunction

Afunction BA:f issaidtobeonetooneorbijection if it isboth

oneoneandonto.

Forexample,if RR:f isdefinedby

i) ( ) 3x2xf + = ii) ( ) 3xxf = forevery Rx thenf isonetoonefunctions.



1.4.12Inverseimageofanelement

Let BA:f bea functionand By . Then the inverse imageof y

under f denotedby ( ) yf 1 - isthesetofthoseelementsofAwhichhaveyastheirimage.

Thatis, ( ) ( ) { } yxf:Axyf 1 = = -

1.4.13Example:.If RR:f isdefinedby ( ) 5x3xxf 2 + - = find(i) ( ) 3f 1 - and (ii) ( ) 15f 1 -

Solution:

i) Let ( ) y3f 1 = - then ( ) 3yf = i.e., 35y3y3 = + - or 02y3y2 = + -

i.e., ( ) ( ) 02y1y = - - Therefore 2yor1y = =

Therefore ( ) { } 2,13f 1 = -

ii) Let ( ) y15f 1 = - Hence ( ) 15yf = Therefore 155y3y2 = + -

010y3y2 = - -

( )( ) 2yor,5yTherefore02y5y - = = = + - Hence ( ) { } 5,215f 1 - = -

1.4.14Inversefunctions

Ifafunction BA:f isoneoneandontothentheinverseoffdenoted

by AB:f 1 - isdefinedby ( ) ( ) { } fy,x:x,yf 1 = -

Thus if BA:f is both one one and onto then AB:f 1 - is

obtainedbyreversingtheorderedpairsoff.



Notethatf1existsonlywhen f isbothoneoneandonto.Furtherf1 is

alsooneoneandonto.

1.4.15Example:LetQbethesetoftherationals.If QQ:f isdefinedby

f(x)=2x3forevery Qx thenfindf1 ifitexists.

Solution:

i) Let f(x1)=f(x2)

2121 xx3x23x2 = - = -

Hence f isoneone.

ii) Let Qy .Thentofind ( ) yxf:Qx =

i.e.,2

3yxThereforey3x2

+ = = -

Whenever y is rational,2

3yx + = is also a rational. Hence there

exists Q2

3y

+ suchthat y

2

3yf =

+

Hence f isonto.Therefore QQ:f 1 - exists.

Letx=f1(y) Thereforey=f(x)

i.e.,2

3yxor3x2y

+ = - =

Define QQ:f 1 - by ( ) 2

3yyf 1

+ = - forevery Qy .

Replacingybyx,weget ( ) Qx2

3xxf 1

+ = - .

Thisisrequiredinversefunction.



1.4.16CompositefunctionorProductfunction

If BA:f and CB:g aretwofunctionsthenthecompositefunction

off and gdenotedbygofisafunctionfromA toCdefinedby

( ) ( ) ( ) { } xfgxfog = forevery Ax Here ( ) ( ) ( ) { } ( ) zygxfgxfog = = =

1.4.17Example:LetRbethesetofrealnumbers.Define

RR:gandRR:f byf(x)=3x2and ( ) 4xxg 2 + = .Find(i)gof (ii) fog

Solution

i) ( ) ( ) ( ) { } ( ) 2x3gxfgxfgo - = = ( ) 42x3 2 + - =

8x12x9 2 + - = forevery Rx

( ) xfy =

( ) ygz =

fog

x

A B C

g

Figure1.1



ii) ( ) ( ) ( ) { }

+ = = 4xfxgfxogf 2

24x3 2 -

+ =

10x3 2 + = forevery Rx

Note that foggof .Thatisthecompositionofmappingsisnotcommutative.

1.4.18Example: If ( ) 1x,x1

1xf -

= findf[f{f(x)}]

Solution

Now ( ) { } x1

11

1

x1

1fxff

- -

=

- =

x1x

xx1 -

= - -

=

Therefore ( ) { } [ ] ( )

x1xx

x

x

1x1

1x

1xfxfff =

- - =

- -

=

- =

Hencef[f{f(x)}]=x

1.5Intervals

Let Rb,a suchthat ba < .Thesetofrealnumbersx suchthat

a



Further,semiopenorsemiclosedintervalsaredefinedasbelow

[a,b)={x R:a x



1.6.1Example: If f is realvaluedfunction find thedomainof the following

functions.

(i) ( ) 6x5x

xxf

2 + - = (ii) ( ) 2x9xf - = (iii) ( )

2x1

xxf +

=

Solution:

Thedomainoffunctionfisthesetofallrealnumbersxforwhich f(x)isa

realnumberi.e., f(x) ismeaningful.

i) Thefunction ( ) 6x5x

xxf

2 + - = isdefinedforallrealvaluesofx except

when

( ) ( ) 3x,2xor03x2xor06x5x2 = = = - - = + - .Hence f(x)isnotdefinedforx=2and

x=3.Hencedomain(f)=R{2,3}

ii) Clearly ( ) 2x9xf - = is defined for all real x for which thequantityundertheradicalsigni.e., 2x9 - ispositivei.e., 0x9 2 -

or ( ) ( ) 3x30x3x3 - + - .Hencedomainoff(x)={x:3 x 3}=[3,3]

iii) Clearly, ( ) 2x1

xxf +

= isdefinedforallrealvaluesof x.Sodomain

of f(x)=R

1.6.2Example:Findtherangeofthefollowingrealfunctions

(i) ( ) 9x

1xf -

= (ii) ( ) 2xxf = (iii) ( ) x1

xxf -

=



Solution:

Sincerangeof f={y:f(x)=yandx domainf},wesubstitutef(x)=yand

expressxintermsofyintheform ( ) yx F = ,say,andsolveforyforwhichthevaluesofxareinthedomainoff.

i) Let ( ) y

1y9xy

9x1xf

+ = =

- =

Clearly,xisnotdefinedwheny=0

Hencerangeof f=R{0}

ii) Let yxyx)x(f 2 = = =

Clearly,xisdefinedforallpositiverealvaluesofy.Hencerangeoffis

thesetofallnonnegativerealnumbersi.e.,[0, ).

iii) Let ( ) y1

yxy

x1xxf

+ = =

- =

Clearly, x is defined for all real values of y except when 1+y = 0

i.e.,wheny=1

Hencerangeoff=R{1}

1.6.3Example:Findthedomainandrangeofthefunction2x1

x)x(f +

=

Solution:

Domain: Clearly, f(x)isdefinedforallrealvaluesofx.

Hencedomainof f=R

Range: Let 0yxyxyx1

xy)x(f 22

= + - = +

=

y2

y411x

2 - = whichisarealnumberif14y2 0and 0y

01y4 2 - and 0y



041y2 - and 0y

021y

21y

+

- and 0y

21y

21 - and 0y

-

2

1,00,

2

1y

Forx=0wehavey=0

Hence,rangeof

- =

2

1,00,

2

1f

1.6.4Example:Findthedomainandrangeoftherealfunction

( ) x3cos2

1xf -

=

Solution:

Giventhat ( ) x3cos2

1xf -

=

Domain:since q cos liesbetween1and+1,

i.e., ,1cos1 - q wehave

,1x3cos1 - for x3 = q

( ) 12x3cos212 - - - - 1x3cos23 -

3x3cos21 - forallrealx

Hence f(x)isdefinedforall Rx

Hencedomainof f=R



Range:Let ( ) y1x3cos2y

x3cos21yxf = - =

- =

y12x3cos - =

But 1x3cos1 -

1y

121Therefore - -

21y121 - - - -

1y13 - - -

1y13

1y31

1,

31y

So,rangeof

= 1,

3

1f

1.7Differentfunctions

In thisarticlewe shall discuss somestandard real valued functionswhich

areofmuchimportanceinthestudyofcalculus.

1.7.1 Periodic function: A function BA:f is said to be periodic if

( ) ( ) a + = xfxf forevery Ax andforsome R a .Here a iscalledtheperiodofthefunction.



For example: f(x) = sin x is a periodic function with period 2p since

sin(2 p +x)=sinxf(x)=tanxisaperiodicfunctionwithperiod p sincetan

(p +x)=tanx.

1.7.2 Algebraic function: the function y = f(x) which consists of finite

numberoftermsinvolvingpowerandrootsoftheindependentvariablex,is

calledanAlgebraicfunction.Ifradicalsignsorfractionalindicesoccurinthe

function,itissaidtobeirrationalotherwiseitissaidtoberational.

Forexample:

( ) 1xxy,1x2y

,6x5x

3x2y,8xy,1xxy

23

1

232

+ + = + =

+ -

+ = + = + - =

,

are all algebraic functions the former three functions are rational algebraic

functionswhereasthelattertwoareirrationalalgebraicfunctions.

AfunctionwhichisnotalgebraiciscalledTranscendentalfunction.

Forexample: xlog,e,xsin x aretranscendentalfunctions.

1.7.3Trigonometricfunctions:Thesefunctionsaredefinedasbelow.

Sine function: The function RR:sin where R is the set of real

numbersdefinedbysinx=yforevery Rx iscalledSinefunction.Here

Rx aretheradianmeasuresoftheangle.

Inotherwords, thefunctionthatassociateseachrealnumberx tosinx is

calledthesinefunction.

ThedomainofthesinefunctionisRandtherangeis[1,+1]

Similarly, the other trigonometric functions viz., cosine, tangent, cosecant,

secantandcotangentcanbedefined.Howeverthedomainandrangeofthe

trigonometricfunctionsareasstatedbelow:



Function x domain y range definitionofthefunction

sin R [ ] 1,1 + - yxsin = cos R [ ] 1,1 + - cosx=ytan ( )

+ - Zn:2

1n2R p R yxtan =

cosec { } Zn:nR - p (,1] [1, ) yxcosec = sec ( )

+ - Zn:2

1n2R p (,1] [1, ) secx=y

Cot { } Zn:nR - p R cotx=y

Note that tanfunctionandsecfunctionarenotdefinedatoddmultiplesof

2 p whereas cosec function and cot function are not defined at

Zn,2

.n2nx = = p p i.e.,atevenmultiplesof2 p .

1.7.4 Inverse Trigonometric functions: We know that the inverse of

a function f exists only when f is both one one and onto i.e., f is

bijective. The sine function RR:sin defined by sin x = y for every

Rx is not oneone since sine function is periodical i.e.,

( ) xsinxn2sin = + p wheren isan integer.However, ifwe restrict the

domain of the function to the set

2

,2

_ p p and the corresponding

rangetotheset[1,1],thefunction [ ] 1,12

,2

:sin + -

+ -

p p

definedby

+ - " =

2,

2xyxsin p p

isbothoneoneandonto(Provethis)



Thedomainsandtherangesforwhichthetrigonometricfunctionsareboth

oneoneandontoaregivenbelow:

Function x domain y range definitionofthefunction

sin

+ -

2,

2 p p [1, +1] yxsin =

cos [0, p] [1, +1] yxcos =

tan

+ -

2,

2 p p R yxtan =

cosec { } 02

,2

-

+ - p p ( ) 1,1R - - yxcosec =

sec [ ]

- 2

,0 p p ( ) 1,1R - - yxsec =

Cot (0, p) R yxcot =

Note that tan p p p

cot,0cotand2

sec0,cosec,2

are not defined.

Wenowdefinetheinversetrigonometricfunctionsasfollows.

(i) Inverseofthesinefunction

Inverseofthesinefunctiondenotedby 1sin - isthefunction

[ ]

- + - -

2,

21,1:sin 1

p p definedby

xysin 1 = - forevery y [1,+1]

Whichisalsobothoneoneandonto.Thus ysin 1 - istheanglebetween

2

p - and

2 p whosesineisy.Thisiscalledtheprincipalvalueof ysin 1 - .



Forexample,since

- =

223and

2

3

3sin

p p p p ,wehave

323

sin 1 p = -

Whereas,

- =

2,

23

2and

2

3

3

2sin

p p p p ,

Hence32

23

sin 1 p -

(ii)Inverseofthecosinefunction

Inverse of the cosine function denoted by 1cos - is the function

[ ] [ ] p ,01,1:cos 1 + - - definedby xycos 1 = - forevery [ ] 1,1y + -

Thefunction [ ] p ,0ycos 1 - iscalled theprincipalvalueof inverseofthecosinefunction.

Forexample,since [ ] p p p ,03

and21

3cos = ,wehave

321cos 1 p = -

Also [ ] 32

21cos,0

32and

21

32cos 1 p p p p =

- - = - etc.

(iii)Inverseofthetangentfunction

Inverse of the tangent function denoted by 1tan - is the function

- -

2,

2R:tan 1 p p definedby

xytan 1 = - forever Ry

Thefunction

- -

2,

2ytan 1 p p istheprincipalvalueof 1tan - function,



Forexample,since

- =

2,

23,3

3tan p p p p wehave

33tan 1 p = -

Also, ( ) 4

1tan 1 p - = - - since

- -

2,

24 p p p

Similarly, 111 cotandsec,cosec - - - functionscanbedefined.However,

thedomainandrangeofthesefunctionsareasstatedbelow:

i) The domain of 1eccos - is ( ) 1,1R - - , while its range is

{ } 02

,2

-

- p p

ii) Thedomainof 1sec - is ( ) 1,1R - - ,whileitsrangeis [ ] 2

,0 p p -

iii) Thedomainof 1cot - isR,whileitsrangeis ( ) p ,0(iv)Notation

i) The functions xtan,xcos,xsin 111 - - - etc. are also denoted by

arcsinx,arccosx,arctanxetc.

ii) xsin 1 - shouldnotbeconfusedas ( ) xsin

1xsin 1 = - etc.

1.7.5Exponentialfunction

Let + R bethesetofpositiverealnumbers.Afunction + RR:f defined

by ( ) 0a,axf x > = is called the exponential function. In particular, ifea = where 71828.2e = approx, then ( ) xexf = is the exponential

function.



1.7.6Logarithmicfunction

Afunction RR:f + definedby ( ) xlogxf a = ,where + R1a iscalled a logarithmic function. In particular if ea = , then ( ) xlogxf e = iscalledthenaturallogarithm

1.7.7HyperbolicFunctions:

Hyperbolic functions are defined in termsof the exponential functions xe

and xe - asfollows

1.2ee

xsinhxx - -

= 2.2ee

xcoshxx - +

=

3.xx

xx

ee

ee

xhcos

xhsinxtanh

-

-

+

- = = 4.

xx ee

2

xhsin

1xechcos

- - = =

5.xx ee

2

xhcos

1xhsec

- + = = 6.

xx

xx

ee

eexhsinxhcos

xhcot -

-

-

+ = =

Heresinhxiscalledhyperbolicsinefunctioncoshxiscalledthehyperbolic

cosinefunctionetc.

Byusingtheabovedefinitionswecaneasilyprovethefollowingidentities.

1. 1xhsinxcosh 22 = -

2. xhtan1xhsec 22 - =

3. 1xhcotxhcosec 22 - =

4. ( ) yhsinxhcosycoshxhsinyxsinh = 5. ( ) ysinhxhsinycoshxcoshyxcosh = 6. ( )

yhtanxhtan1yhtanxhtan

yxtanh

=

7. xcoshxhsin2x2sinh =



8. xsinh211xcosh2xsinhxcoshx2cosh 2222 + = - = + =

9.xhtan1

xhtan2x2tanh

2 + =

1.7.8Parametricfunctions

Ifthetwovariablesxandyareexpressedasfunctionofathirdvariabletby

x=f(t),y=g(t),thensuchfunctionsarecalledparametricfunctions.The

thirdvariabletiscalledtheparameter.

For example, consider the function y2 = 4ax. This function is satisfied by

( ) 222 at.a4at2forat2yandatx = = = i.e., 2222 ta4ta4 = .Hencewe say that at2y,atx 2 = = are the parametric functions of the

givenfunction ax4y2 =

Similarly, q q sinay,cosax = = are the parametric functions of

222 ayx = + .

1.7.9ModulusFunction

Thefunctiondefinedby

( )

= < -

> = =

0xwhen0

0xwhenx

0xwhenx

xxf

iscalledthemodulusfunction.

Thus|x|istheabsolutevalueofx.Forexample,

00,5555 = = - = .

Thedomainof|x|isthesetRofrealnumberswhileitsrangeisthesetofall

nonnegativerealnumbers.



1.7.10TheGreatestIntegerFunction

Let R be the set of reals and Z the set of integers. Then the function

ZR:f definedby

f(x)=[x]forevery Rx

Where[x]denotesthegreatest integer lessthanorequaltoxiscalledthe

greatestintegerfunction.

Notethatthedomainof[x] is Rwhileitsrangeisthesetofallintegersasit

attainsonlyintegralvalues.

Forexample,

[3.75]=3, [2.25]=2,[1.75]=1,[0.25]=0

[3.5]=4,[2.75]=3,[1,25]=2,[0.5]=1

Graphsof ex,logx,|x|and[x]:

We now sketch the graphs of exponential, logarithmic, modulus and the

greatestintegerfunctionsasexplainedbelow.

(i)Graphofex:Thecurvewhoseequationis xey = where 71828.2e =

approximatelyiscalledexponentialcurve

Thiscurvepassesthroughthepoint(0,1)whichliesabovethexaxisand

hasxaxisasasymptote(Anasymptoteisthestraightlinewhichtouchesa

curveat infinity,butwhich isnotaltogetherat infinity).That is thecurve is

extending towards but never touches the x axis. The shape of the

curvemaybeobservedinthegraphintheFigure1.2.

Tosketchthegraphof xey = ,weassigntheconvenient(smaller)values

toxandfindthecorrespondingvaluesof ybytaking 71828.2e = whose

4343.0elog10 = ,



y=ex

x 4 3 2 1 0 1 2 3 4

y 0.02 0.05 0.13 0.37 1 2.7 7.4 20.1 54.6

Tocomputey whenx=2,

Wehave xey = or ( ) 8686.04343.2elog2ylog = = = Therefore ( ) 4.7389.78686.0ALy = = = Whenx=2,

2ey - = or ( ) 8686.04343.02elog2ylog - = - = - = Therefore ( ) ( ) 13.01314.1AL8686.0ALy = = - =

(ii)Graphoflogx

The curve whose equation is 1a,xlogy a > = is called a logarithmic

curve.Thiscurvepassedthroughthepoint(1,0)whichliestotherightofthe

yaxixandhastheyaxisasasymptote.

Figure1.2

X

Asymptote

60

50

40

30

20

10

1 2 431234

Y

O



Wenowsketchthegraphoflogx, asexplainedbelow.

Example: Sketchthegraphofy=logx.

x 10 5 4 3 2 1 0.5 0.25 0.1 0.01

y=logx 1 0.7 0.6 0.5 0.3 0 0.3 0.6 1 2

Figure1.3

(iii)Graphof|x|

Thecurvewhoseequationisy=|x| iscalledamoduluscurve.Bydefinition

ofthemodulusfunction

Wehave

< -

= =

0xwhenx

0xwhenxxy

Wenowformthetablefordifferentvaluesofx

x 4 3 2 1 0 1 2 3 4

y 4 3 2 1 0 1 2 3 4

X

Y

Asymptote3

3

2

1

1

2

O1041 2 3 5 7 96 8



(iv)Graphof [x]

Thecurvewhoseequationisy=[x]iscalledthegreatestintegercurve.By,

definition,wehaveforanyrealnumber,[x]isthegreatestintegerlessthan

orequaltox.Herethedomain[x]=Randrange[x]=Z

x 2.75 1.5 0.5 0 1.25 2.75 3.5

Y=[x] 3 2 1 0 1 2 3

Y

3

3

21

Figure1.4

12O

41 2 3

4

4X

2

f(x)=x f(x)

=x

Figure1.5

3

3

2

1

12

O

1 2 3

2

1

3




1. Findxandyif(3x+y,x1)=(x+3,2y2x)

2. IfA={1,2,3},B={2,4,5}find

(A B)x(AB)

(b) Ax(AB)

(c)(A D B)x(A B)

3. IfA={x/x Nandx



1.9TerminalQuestions:

1. Defineanequivalencerelationwithasuitableexample.

2. Defineamodulusfunctionwithexample.

1.10Answers


1) TheorderedPairsareequalif3x+y=x+3andx1=2y2x

i.e.2x+y=3

3x2y=1

Solvingx=1,y=1

2) A B={2}

AB={1,3}

BA={4,5}

A D B=(AB) (BA)={1,3,4,5}

(A B)x(AB)=(2)x{1,3}={(2,1),(2,3)}

Ax(AB)={1,2,3}x(1,3)={(1,1),(1,3),(2,1),(2,3),(3,1),(3,3)}

(A D B)X(A B)={1,3,4,5}X{2}={(1,2),(3,2),(4,2),(5,2)}

3) Sincex216=0

A={1,2}andB={4}

(x4)(x+4)=0

BxA={4}x{1,2}

={(4,1),(4,2)} ==>x=4,4

Thereforex=4(x



5) x25x+6=0==>(x2)(x3)=0==>x=2,3

A={2,3},B={2,4}andC={4,5}

AB={3}andBC={2}

Therefore(AB)x(BC)=(3)x(2)=(3,2)

6) (A B)={3,4},(B C)={4}

(A B) C={3,4} {1,4,7,8}={4}

A (B C)={1,2,3,4} {4}={4}

ThereforeA B C=(A B) C=A (B C)

TerminalQuestions

1. RefertoSection1.3.7

2. RefertoSection1.7.9

mc0063(a) unit1 final

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