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1、9AdvancedMicroeconomicsTopic1:Set,Topology,RealAnalysisandOptimization(withthecomplementofAssociateProf.MichaelLi)Readings:JR-Chapters1&2,supplementedbyDL-AppendixC1.1IntroductionInthislecture,wewillquicklygothroughsomebasicmathematicalconceptsandtoolsthatwillbeusedthrough
2、outtherestofthecourse.Asthisisareviewsession,theattentionwillbemainlyonrefreshingonthelanguage,styleandrigorofmathematicalreasoning.Ineconomicanalysis,especiallymicroeconomicanalysis,mathematicsisalwaystreatedasatool,nevertheend.Ontheotherhand,byintegratingeconomicswithrig
3、orousmathematics,wewillbeabletodevelopthetheoreticalexpositionsinasoundandlogicalmanner,whichiswhyeconomicsisalsoknownaseconomicscience.Notmanyothertraditionallyknownassocialsciencefieldsmanagetopassthiscriticalstage.Butitisimportanttorememberthatasaneconomist,wemustgobeyo
4、ndthenormalmathematicaltreatmentandtheunderlyingeconomicsandtheirpolicyimplicationsarefarmoreimportantandinteresting.Theplanofthislecturegoeslikethis.First,wewillreviewthebasicsettheory.Wethenmoveontoabitoftopology.Afterreviewingbasicelementsofrealanalysis,wewillcoversomek
5、eyresultsinoptimization.1.2BasicsofSetTheory1.2.1BasicConcepts·set:acollectionofelements·setsoperations:union,intersection·realsets:(thenotionofvectors)·":forany;$:thereexists;':suchthat;Î:belongsto;isanelementof1.2.2Convexity&RelationsConvexSet:·AsetSÌRnisconvexif·Intuiti
6、vely,asetisaconvexsetifandonlyif(iff)wecanconnectanytwopointsinastraightlinethatliesentirelywithintheset.·Convexsethasnoholes,nobreaks,noawkwardcurvaturesontheboundaries;theyareconsideredas“nicesets”.·Theintersectionoftwoconvexsetsremainsconvex.Relations·Foranytwogivensets
7、,SandT,abinaryrelationRbetweenSandTisacollectionoforderedpairs(s,t)withsÎSandtÎT.·ItisclearthatRisasubsetofS´T:(s,t)ÎRorsRt.99PropertiesofRelations:·Completeness·RÌS´Siscompleteiffforallxandy(x¹y)inS,xRyoryRx.·Reflexivity·RÌS´SisreflexiveifforallxinS,xRx.·Transitivity·RÌS´
8、Sistransitiveifforallx,y,zinS,xRyandyRzimpliesxRz.1.3TopologyTopologyattemptstostudythefu
