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1、MathematicalPreliminaries1.1.Sets•Asetisawell-definedcollectionofelements・Example:A={a,b,c}isasetwiththreeelements・agAmeansthattheelementaisinthesetAdgAmeansthattheelementdisnotinthesetA.•0denotesthenullsetoremptyset,i.e.,thesetwithnoelement.•setAisasubsetofsetB讦awBforallawA.Th
2、isisdenotedbyAuB,orBA・•unionofsets:AuB={x:xgAorxgB}.•intersectionofsets:AnB={x:xgAandxgB}.•AandBaredisjointsetsifAnB=0.•thecomplementofsetAinsetBisthesetBA={x:xgBandxGA}・•Cartesianproduct:LetxgXandyeY,andlet(x,y)beanorderedpair.Then(x,y)6XxY,whereXxYistheCartesianproductofXand
3、Y・1.2.LogicConsidertwopropositionsPandQ・IfPimpliesQ、thenPisasufficientcoixlitionforQ,andQisanecessaryconditionforP.ThisisdenotedbyP=>Q.IfPimpliesQandQimpliesP,then“Pholdsifandonly讦Qholds，：PandQareequivalent,andPisanecessaryandsufficientconditionforQ.ThisisdenotedbyP<=>Q.Let{not
4、P}denotethestatementthatPisnottrue.CorHrapositive:IfPimpliesQ,then{notQ}implies{notP}.Example1:Letxbearealnumber,P={x>0)andQ={x2>0).Then,P=>Qand{notQ}=>{notP}hold,but{Q=>P}isfalse.Example2:LetP={x>0)andQ={x3>0}・ThenPoQ,and{notP)<=>{notQ)hold.1.3.Numbers•naturalnumbers:N={1,2,3,
5、…}・integers:Z={-3,-2,-1,(),L2,3,...}•rationalnumbers:Q={a/b:agZ,beZ,bHO}・irrationalnumbers:2：3…•realnumbers:R={x:xisrationalorirrational}.complexnumbers:C={a+bi:aeR,bgR,i=(-1)12),whereaistherealpart,andbistheimaginarypart・then-foldCartesianproductofR:Rn=Rx...xR={(xhx2,•••,xn):X
6、jgR,i=1,n),whereXjisthei-thcoordinateofx=(xbx2,xn)1.4.FunctionsLetXandYbesets.Definition:f:XtYisamappingassociatingeachelementofXwithanelementofY.Xisthedomainofff(x)istheimageofxunderff(X)={f(x):xgX}istheimageofXunderf•afunction:ifonlyonepointinYisassociatedwitheachpointinX・aco
7、irespondence:ifmorethanonepointinYcanbeassociatedwitheachpointinX•inversefunction:x=f*(y)ifandonlyify=f(x).afunctionf:XTYisonto讦f(X)=Y.Itmeansthattheequationf(x)=yhasatleastonesolutionforeachy.•iff(x)andfly)arebothsingle・valued,thenfison—to-one.Itmeansthattheequationf(x)=yhasat
8、mostonesolutionforeachy.•compositefunction:h=g(f(x))=g
