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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.
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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.
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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).
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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.
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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.
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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 =
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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 =
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(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 = = - -
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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}
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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
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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
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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
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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
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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.
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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.
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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.
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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.
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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.
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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.
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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
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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
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Further,semiopenorsemiclosedintervalsaredefinedasbelow
[a,b)={x R:a x
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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 -
=
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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
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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
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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.
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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:
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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)
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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 - .
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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,
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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.
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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 =
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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.
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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 = ,
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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
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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
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(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
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SelfAssessmentQuestions
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
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1.9TerminalQuestions:
1. Defineanequivalencerelationwithasuitableexample.
2. Defineamodulusfunctionwithexample.
1.10Answers
SelfAssessmentQuestions
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
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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