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1、ElementsofMathematicalStyleAworkinprogress1BruceFrancisDepartmentofElectricalandComputerEngineeringUniversityofTorontoJuly8,20131IthankMurrayWonhamandAudreyKerteszforreadingthisworkando eringhelpfulcomments.ContentsPreface21Preliminaries32De nitionStatements63TheoremStatements114TheElement
2、sofStyle155GivingaTalkUsingComputerSlides201PrefaceThisisaverybriefhandbookonwritingdocumentswithmathematicalcontent:reports,theses,journalpapers,andconferencepapers.Thehandbookiswrittenprimarilyforgraduatestudentsincontrolandsimilarsubjects.Towriteyourtheoremscorrectly,youneedtoknowelemen
3、tarylogic,namely,propositionalcalculus.Themainpointisthatvariables(",,etc.)mustbequanti ed(thereexists,"forall")andthequanti ersmustbeintherightorder.Idecidedtowritethishandbookafteryears
4、infactdecades
5、ofreadingandcorrectingtheses.Nowwhenstudentsaskmetobeontheirthesisexaminationcommitte
6、es,Itellthemtogoandreadthishandbookbeforegivingmetheirtheses.Itreducestheamountofredink.2Chapter1Preliminaries1.Set.Examplesofsets:thesetRofrealnumbers;thesetRmnofmnrealmatrices;thesetF(U;V)offunctionsfromthesetUtothesetV.Asetisfrequentlyde nedusingnotationlikethis:S=fx:x2R;x>0g:Inwordst
7、hissays,Sisthesetofallxsuchthatxbelongstotherealnumbersandxispositive.Thusacolonmeanssuchthat,"acommameansand,"and2meansbelongsto"orisamemberof."NoticethatthesymbolxisadummyvariableandthatwehaveexactlythesamesetifwewriteS=f:2R;>0g:So,precedingthecolonisasymbolforavariable,andfollowi
8、ngthecolonaredescriptorsaboutthatvariable.AvariationisS=fx2R:x>0g;whereoneofthedescriptorsofxprecedesthecolon.Anothercommonformofwritingthesetistouseaverticallineinsteadofacolon:S=fxjx2R;x>0g:Supposeyouwantthesetofrealnumbersgreaterthananumbera.YoucouldwriteLeta2Randde neS=fx:x2R;x>ag:Oryo
9、ucouldwriteDe neS=fx:x2R;x>ag;a2R:3CHAPTER1.PRELIMINARIES4Inwords:Sisthesetofallxsuchthatxbelongstotherealnumbersandxisgreaterthana,whereabelongstothesetofrealnumbers.Butthisiswrong:S=fx:x2R;x>a;a2Rg:Infact,thislattersetisallofR,foritsaysSisthesetofallxsuchtha
