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1、Chapter11、proofLetA,B,Cbesets.Supposethatx∈B,wegetx∈A∩Bor,andorsinceand.sox∈Cand.Similarly,wehaveandsoB=C.2、proof①First,consider.Thenor,but.ThisimpliesifxisnotanelementofA,then.Henceand.Conversely,if,thenbydefinition,or.Thisgeneratestwocases:(a1)If,clearly;(b2)If,theneitherornot.i.e.,
2、eitherandorbut,ineithercase,wehave.Hence.Therefore=.②Supposethat.Thenbutand.So-Bandandbydefinition.Hence.Converssely,Assumethat,then-Band,andwehavebutand.Hence,,i.e.,.Thereforeand,so=3.(a)surjective(b)bijective(c)bijective4.proofiff:XYandg:YZarefunctions,thentheircompositedenotedbygf,
3、isthefunctionXZgivenbygf:Xg(f(x))(i)supposethat(gf)(a)=(gf)(b),wherea,bX.wehaveg(f(a))=g(f(b))bydefinition,andf(a)=f(b)sincegisinjective,similarly,a=bforfisinjective.Therefore,gfisinjective.(ii)ForeachZZ,thereisyYwithg(y)=zsincegissurjective,andforeachyY,thereexistsaxwithf(a)=ysincefi
4、ssurjective.SoforzZ,thereisaxwith(gf)(a)=g(f(a))=g(y)=z.whichimpliesgfissurgective.5.proofclearly,:RRisafunction.Supposethat(a)=(b)wherea,bRaredistinct.Then,crossmultiplyingyields,whichsimplifiestoandhence,soisinjective.forgivenyR,from,wegetequation,whichcanbesolvedforx,i.e.foreachyR,
5、thereisatleastxxsuchthat.whicimpliesissurjective.Thereforeisbijective.6、(a)Risreflexive,symmetric,transitive.(b)Risreflexive,notsymmetric,transitive.(c)Risreflexive,symmetric,transitive.(d)Risreflexive,symmetric,transitive.7、proof(1)Forevery∈R-{0}，wehave,andso(2)If,where,i.e.,then,i.e
6、.,,(3)If,where,i.e.,then.i.e.,.Therefore,therelation～isanequivalencerelation.8、Thereare1,3,5,15equivalencerelationsonasetSwith1,2,3or4elements,separately.9、Wecanlisttheelementsoftheresidueclassesofmodulo3:[0]={…,-9,-6,-3,0,6,9,…}[1]={…,-8,-5,-2,1,4,7,10,…}[2]={…，-7,-4,-1,2,5,8,11,…}Ch
7、apter21、proofi)ii）Foreveryx,y,z∈G,(x*y)*z=(xy-x-y+2)*z=(xy-x-y+2)z-z-(xy-x-y+2)+2=xyz-xz-yz+z-xy+x+yx*(y*z)=x*(yz-y-z+2)=x(yz-y-z+2)-x-(yz-y-z+2)+2=xyz-xy-xz+x-yz+y+zwehave(x*y)*z=x*(y*z).Andsotheassociativelawholds.3、SolutionStraightforwardcalculationshowsthat.,since.4、proofSupposefo
8、rall，
