[PDF] Module A-5: Injective Surjective and Bijective Functions





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Is a function injective or surjective?

    A function is injective (an injection or one-to-one) if every element of the codomain is the image of at most one element from the domain. A function is surjective (a surjection or onto) if every element of the codomain is the image of at least one element from the domain. A bijection is a function which is both an injection and surjection.

What is the difference between surjective and injective?

    Surjective: If f: P ? Q is a surjective function, for every element in Q, there is at least one element in P, that is, f (p) = q. Injective: If f: P ? Q is an injective function, then distinct elements of P will be mapped to distinct elements of Q, such that p=q whenever f (p) = f (q).

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What is injectivity in math?

    Recap: Injectivity ?A function is injective(one-to-one) if every element in the domain has a unique image in the codomain –That is, f(x) = f(y) implies x= y NY MA CA Albany Sacramento Boston ...
ModuleA-5:Inject ive,Surject ive,andBijectiveFunctions

Math-270:DiscreteMat hematics

November10,2019

Motivation

You'resurelyfami liarwiththeideaofani nversefunction:afun ctionthatundo essomeother function.For example, f(x)=x 3 andg(x)= 3 p x areinv ersesofeachother.Whetherth ink ingmathemat icallyorcodingth isinsoftware,thingsgetcom pli- cated. Thetheoryof injective,surj ect ive,andbijectivefunctionsisaver ycompactandmostlystraightforward theory.Yetitcomplet elyuntangle sallth epotentialpitfallsofinvertinga function.

Terminology

Ifafun cti onfmapsasetXtoas etY,w eareaccu stomedt ocallingXthedomain( whichisfine)b utwe arealsoac customedt ocallingYtherange,an dthatissloppy. Theran geoffisthe setofvalues actually hitby f.I notherw ords,yisint herangeof f(x)if andonlyi fthere issomexinthe domainsucht hat f(x)=y.Wi thoutthisrestricti on,werefertoYasth eco-domainoff(x). Youhave probablyheardth ephrase"yisthe imageofx"whenf(x)=y.Lik ewise,wecansay"xisa pre-imageofy."Notice thatwesay"apr e-image"andn ot"thep re-image."T hat' sbecauseymighthave multiplepre-images.

Forexam ple,iff(x)=x

2 asaf unc tionoftherealline,theny=4 hast wopre- images:x=2an d x=2.Me anwhile,y=0 hason lyonepr e-image,x=0.I ncon trast, y=1has nopre -images.

InjectiveFunctions

FormalDefintion: Afu nctionf:D!Cisinje ctiveifandonlyif "forallx 1

2Dandx

2

2Diff(x

1 )=f(x 2 )thenx 1 =x 2 CasualDefiniti on:Notw odistinct pointsinthedomainmaptothesam evalue.

ClassicExample:f(x)=e

x ,th oughtofasf:R!R. HorizontalLineTest:Everyhorizontalli nehitsthecurveatmos tonce.

EasyNon-Exam ple:f(x)=sinx,th oughtofasf:R!R.

Pre-images:Everypointinthec o-domainhasatmost onepre- image.

SurjectiveFunctions

FormalDefintion: Afu nctionf:D!Cissurj ectiveifandonlyif 1 CasualDefiniti on:Everypointinthec o-domainhassomep ointint hedomainthat mapstoit.

ClassicExample:f(x)=t anx,t houghtofasR

5⇡

2

3⇡

2 2 2

3⇡

2

5⇡

2 !R. HorizontalLineTest:Everyhorizontallin ehitsthecurveatle astonce.

EasyNon-Examp le:f(x)=e

x ,th oughtofasf:R!R. Pre-images:Everypointinthec o-domainhasatleas tonepre -image.

BijectiveFunctions

FormalDefintion: Afu nctionfisbije ctiveifandonlyifitisbothinjec tivean dsurje ctive. CasualDefinition :Everypointinthec o-domainhasexact lyonepoi ntinthedom ainthatmapstoit.

ClassicExample:f(x)=x

3 ,th oughtofasR!R. HorizontalLineTest:Everyhorizontallin ehitsthecurveexactlyonce.

EasyNon-Exam ple:f(x)=x

2 ,th oughtofasf:R!R. Pre-images:Everypointinthec o-domainhasexact lyonepr e-image.

MathematicalTerminologyUsedinOthe rTextbooks

•Injectivefunctionsaresometime scalled"injections,"whichi sfine. •Surjectivefunctionsaresometimes called"surjections,"whichi sfine. •Bijectivefunctionsareoftencal led"bijections,"whichisfine . •In100-lev elcourses,wesometim essay"f(x)is invert ible"insteadof"f(x)is bijec tive,"andthat's okay.Itwouldtr aumatize 100-levelstude ntstolearnaboutthesefinerdi stinctionswhich(att hat level)wouldbeseriou slyconfusing. •Insome textbooks, injectivefunctionsarecalled"one -to-one"functions,especiallyatl owerlevels. However,thisphraseissome timesusedfor bijections,andth erefore ,itshouldbeavoided.Whenyou seethephr ase"one-to-one functions,"itis ambiguous,becausesomeauthor swillusethatphraseto indicateinjectivefunc tions,andsomewillusethatphraset oindicatebijectivefunctions. Asalways , thebestp lanistoavoidamb iguityenti rely, anduset heformalvocabul aryinsteadofmathematical slang. •Insome textbooks, wemightsee"thefunctionf(x)i sonto"in placeof"thef uncti onf(x)issurjective." However,thisispainfulto anearaccustomed toprope rgrammar,andshouldnotb eused. Also the phrases"f(x)map sAontoB"versus"f(x)map sAintoB"are toosi milartoeach other.Thehuman earmight mistakeoneforth eother,buttheformer indicat esasu rjectivef unction,whereast helatter doesnotsayanyt hingabou tsurject ivity.

TheThree FormalDefiniti ons

Hereisare capofthe form aldefinitions, fore aseofrefere nce. •Afu nctionf:D!Cisinje ctiveifandonlyif "forallx 1

2Dandx

2

2Diff(x

1 )=f(x 2 )thenx 1 =x 2 2 •Afu nctionf:D!Cissurj ectiveifandonlyif •Afu nctionfisbije ctiveifandonlyifitisbothinjec tivean dsurje ctive.

HowtoProv eTheseT hings

Surjective

Thebestwa ytoprovethat somefuncti onissu rjectiveistopro videaform ulathat,giv enanyy-valueinthe co-domain,willproduceanx-valueinthedomains uchth atf(x)=y. Inother words,ifsomeon easksyoutoprovet hatsomet hingexists,theb estwayto accompli shthis isto producethatverything. Thenegation oftheformaldefiniti oni sfairlyint eresting. ⇠(8y2C9x2Df(x)=y)

9y2C⇠(9x2Df(x)=y)

9y2C8x2D⇠(f(x)=y)

9y2C8x2Df(x)6=y

Asyou cansee, therec ipe(forprovi ngthatafun ctionisnotsurjective)istolocatesomey-valueinthe co-domain,forwhichthereis nox-valueinthedomainw heref(x)=y.

Injective

Thebestwa ytoprovethat somefuncti onisinj ectiveistousea directpro of. Letx 1

2D,letx

2

2D,an dsuppos ethatf(x

1 )=f(x 2 ).Then youdosomealgebr a,unti lyoureac h x 1 =x 2 .Con clude"iff(x 1 )=f(x 2 )thenxquotesdbs_dbs4.pdfusesText_8
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