离散数学好像不怎么重要逻辑与证明logic

DiscreteDiscreteMathematicsandits—LogicandProofs(4)XiaocongDepartmentofComputerDec. XiaocongZHOUXiaocongZHOUDiscreteDec. 1/ fiers: TheOrder Negating XiaocongZHOU

Discrete

Dec. 2/ fiers: fiersarenestedifoneiswithinthescopeoftheNotethateverythingwithinthescopeofa fiercanbethoughtofasapropositionalfunction weneedtounravelwhat fiersandpredicatesthatappearwhichisthesamethingaswhereQ(x)is∃yP(x,y),whereP(x,y)isx+y=XiaocongZHOU

Discrete

Dec. 3/ fiers: fiersarenestedifoneiswithinthescopeoftheNotethateverythingwithinthescopeofa fiercanbethoughtofasapropositionalfunction weneedtounravelwhat fiersandpredicatesthatappearAssumethat consistsofallrealEveryrealnumberhasanadditiveinverse:∀x∃y(x+y=∀x∀y((x>0)∧(y<0)→(xy<XiaocongZHOU

Discrete

Dec. 3/ fiers: ficationasInworking ficationsofmorethanonevariable,itsometimeshelpfultothinkintermsofnestedweloopthroughthevaluesforx,andforeachxweloopthroughthevaluesforyIfwefindthatP(x,y)istrueforallvaluesforxandy,∀x∀yP(x,y)isIfweeverhitavaluexforwhichwehitavalueyforwhichP(x,y)isfalse,XiaocongZHOU

Discrete

Dec. 4/ fiers: ficationasInworking ficationsofmorethanonevariable,itsometimeshelpfultothinkintermsofnestedweloopthroughthevaluesforx,andforeachxweloopthroughthevaluesforyuntilwefindayforwhichP(x,y)istrueIfforeveryxwehitsuchay,then∀x∃yP(x,y)isIfforsomexweneverhitsuchay,then∀x∃yP(x,y)isXiaocongZHOU

Discrete

Dec. 4/ fiers: ficationasInworking ficationsofmorethanonevariable,itsometimeshelpfultothinkintermsofnestedweloopthroughthevaluesforxuntilwefindanxforwhichP(x,y)isalwaystruewhenweloopthroughallvaluesforyOncewefindsuchanx,weknowthat∃x∀yP(x,y)isIfweneverhitsuchanx,thenweknowthat∃x∀yP(x,y)isXiaocongZHOU

Discrete

Dec. 4/ fiers: ficationasInworking ficationsofmorethanonevariable,itsometimeshelpfultothinkintermsofnestedweloopthroughthevaluesforx,whereforeachxweloopthroughthevaluesforyuntilwehitanxforwhichwehitayforwhichP(x,y)istrue∃x∃yP(x,y)isfalseonlyifweneverhitanxforwhichwehitaysuchthatP(x,isXiaocongZHOU

Discrete

Dec. 4/TheOrder Theorder Itisimportanttonotethattheorderof fiersisunlessall fiersare fiersorallareTheorderofnesteduniversal fiersinastatementwithoutother fierscanbechangedwithoutchangingthemeaningthestatementLetP(x,y)bethestatement“x+y=y+where forallvariablesconsistsofallrealTheorderofnestedexistential fiersinastatementwithoutotheruniversal fierscanbechangingthemeaningtheXiaocongZHOU

Discrete

Dec. 5/TheOrder Theorder Itisimportanttonotethattheorderof fiersisunlessall fiersare fiersorallareTheorderofnesteduniversal fiersinastatementwithother fiersmakesdifferentmeaningsofthe LetQ(x,y)bethestatement“x+y=where forallvariablesconsistsofallrealTheorderofnestedexistential fiersinastatementwithother fiersmakesdifferentmeaningsofthe XiaocongZHOU

Discrete

Dec. 5/TheOrder ficationsoftwoWhenWhenP(x,y)istrueforeverypairx,Thereisapairx,yforwhichP(x,isForeveryxthereisayforP(x,y)isThereisanxsuchthatP(x,y)falseforeveryThereisanxforwhichP(x,y)trueforeveryForeveryxthereisayforP(x,y)isThereisapairx,yforwhichP(x,isP(x,y)isfalseforeverypairx,XiaocongZHOU

Discrete

Dec. 6/TranslatingBetweenNaturalLanguagesandLogicTranslatethestatement“ThesumoftwopositiveintegersisTranslatethestatementEveryrealnumberexceptzerohasamultiplicativeinverse”(除0之外的所有实数均有乘逆)Expressthedefinitionofalimit XiaocongZHOU

Discrete

Dec. 7/TranslatingBetweenNaturalLanguagesandLogicTranslatethestatement“ThesumoftwopositiveintegersisExpressthedefinitionofalimit XiaocongZHOU

Discrete

Dec. 7/TranslatingBetweenNaturalLanguagesandLogicTranslatethestatement“ThesumoftwopositiveintegersisExpressthedefinitionofalimit XiaocongZHOU

Discrete

Dec. 7/TranslatingBetweenNaturalLanguagesandLogic fiersintonaturalC(x)is“xhasacomputer”,F(x,y)is“xandyare forbothxandyconsistsofallF(a,b)meansaandbare forx,yandzconsistsofallXiaocongZHOU

Discrete

Dec. 8/TranslatingBetweenNaturalLanguagesandLogic fiersintonaturalC(x)is“xhasacomputer”,F(x,y)is“xandyare forbothxandyconsistsofallF(a,b)meansaandbare forx,yandzconsistsofallXiaocongZHOU

Discrete

Dec. 8/TranslatingBetweenNaturalLanguagesandLogic“If isfemaleandisaparent,then is “Everyonehasexactlyonebest“Thereisawomanwhohastakenaflightoneveryairlineinthe XiaocongZHOU

Discrete

Dec. 9/Negating Statementsinvolvingnested fierscanbenegatedbysuccessivelyapplyingtherulesfornegatingstatementsinvolvinga fierstoexpressTheredoesnotexistawomanwhohastakenaflightonevery