離散數(shù)學(xué)好像不怎么重要邏輯與證明logic

DiscreteDiscreteMathematicsandits—LogicandProofs(5)XiaocongZHOUDepartmentofComputerScienceSunYat-senJan.XiaocongZHOU Discrete Jan. 1/ RulesofInference: RulesofInferenceinPropositional RulesofInference XiaocongZHOU

Discrete

Jan. 2/

RulesofInference:ProofsinmathematicsarevalidargumentsthatestablishthetruthAnargumentmeansasequenceofstatementsthatendwithaAnargumentisvalidifandonlyifitisimpossibleforallthepremisestotrueandtheconclusiontobePremisesisthestatementswhosetruthswealreadyhaveConclusionisthestatementwhosetruthwewanttoRulesofinferenceare tesforconstructingvalidRulesofinferenceareourbasictoolsforestablishingthetruthofXiaocongZHOU

Discrete

Jan. 3/RulesofInferenceinPropositionalAnargumentinpropositionallogicisasequenceofAllbutthefinalpropositionintheargumentarecalledThefinalpropositioniscalledtheAnargumentisvalidifthetruthofallitspremisesimpliesthatconclusionisAnargumentforminpropositionallogicisasequenceofAnargumentfromisvalidnomatterwhichparticularpropositionsaresubstitutedforthevariablesinitspremisesandthetheconclusionistrueifthepremisesareallTheargumentformwithpremisesp1,p2,···,pnandconclusionqisXiaocongZHOUDiscreteJan. 4/when(p1∧p2∧···∧XiaocongZHOUDiscreteJan. 4/RulesofInferenceinPropositionalRulesofinferenceinpropositionalRulesofinferencearerelativelysimpleargumentformswhosevalidityhavebeenestablishedTheserulesofinferencecanbeusedasbuildingblockstoconstructcomplicatedvalidargumentImportantrulesofinferenceinpropositionalModusponens(分離規(guī)則),i.e.lawof ent:(p∧(p→q))→Modustollens(拒取式(?q(p→q→ ):((p→q)∧(q→r))→(p→ ):((p∧q)∧?p)→Addition(附加律p→(pSimplification(化簡(jiǎn)律(pq→Conjunction(合取律((p(q(pResolution(消解律((pq(?pr(qXiaocongZHOU

Discrete

Jan. 5/RulesofInferenceinPropositionalRulesofinferenceinpropositionalExample(ModusSupposethattheconditionalstatementanditshypothesisareBymodusponens,itfollowsthattheconclusionofthestatementisXiaocongZHOUDiscreteJan.XiaocongZHOUDiscreteJan. 5/RulesofInferenceinPropositionalRulesofinferenceinpropositionalAvalidargumentcanleadtoanincorrectconclusiononormoreofitspremisesisDeterminewhethertheargumentgivenhereisanddeterminewhetheritsconclusionmustbetruebecauseofthevalidityoftheargument“If√2>3then(√2)2>(3 “Weknowthat√2>32“Consequently,(√2)2=2>(3)2=9XiaocongZHOUXiaocongZHOUDiscreteJan. 5/RulesofInferenceinPropositionalRulesofinferenceinpropositionalExample(AdditionStatewhichruleofinferenceisthebasisofthefollowing“Itis 雨Example(SimplificationStatewhichruleofinferenceisthebasisofthefollowing“Itis “Thereforeitis XiaocongZHOU

Discrete

Jan. 5/RulesofInferenceinPropositionalRulesofinferenceinpropositionalExample(HypotheticalStatewhichruleofinferenceisusedinthe“Ifwedonothaveabarbecuetoday,thenwewillhaveaXiaocongZHOUDiscreteJan. 5/XiaocongZHOUDiscreteJan. 5/RulesofInferenceinPropositionalUsingrulesofinferencetobuildShowthatthe“Itisnotsunnythisafternoonanditiscolderthanyesterday(今下午沒(méi)有“Wewillgoswimmingonlyifitissunny(只有 “Ifwedonotgoswimmingthenwewilltakeacanoetrip(如果我們不“Ifwetakeacanoetripthenwewillbehomebysunset(leadtothe“Wewillbehomebysunset(我們?nèi)章鋾r(shí)回家XiaocongZHOU

Discrete

Jan. 6/RulesofInferenceinPropositionalUsingrulesofinferencetobuildShowthatthe“Ifyousendmeane-mailmessagethenIwillfinishwritingtheprogram(“Ifyoudonotsendmeane-mailmessagethenIwillgotosleepearly(“IfIgotosleepearlythenIwillwakeupfellingrefreshed(leadtothe“IfIdonotfinishwritingtheprogram,thenIwillwakeupXiaocongZHOUDiscreteJan. 6/XiaocongZHOUDiscreteJan. 6/

RulesofInferenceinPropositionalComputerprogramshavebeendevelopedtoautomatethetaskreasoningandprovingManyoftheseprogramsmakeuseofaruleofinferenceknownThisruleofinferenceisbasedonthe((p∨q)∧(?p∨r))→(q∨XiaocongZHOUDiscreteJan. 7/ThefinaldisjunctionintheresolutionruleqXiaocongZHOUDiscreteJan. 7/

RulesofInferenceinPropositionalUseresolutiontoshowthatthe“ItissnowingorBart imply“JasmineisskiingorBartis XiaocongZHOU

Discrete

Jan. 7/

RulesofInferenceinPropositional ysanimportantroleinprogramminglanguagesbasedontherulesoflogicsuchasProlog,whereresolutionrules ResolutioncanbeusedtobuildautomatictheoremprovingToconstructproofsinpropositionallogicusingresolutionastheonlyruleofinference,thehypothesesandtheconclusionmustbeexpressedasclausesAclauseisadisjunctionofvariablesornegationsoftheseWecanreceastatementinpropositionallogicthatisnotaclausebyoneormoreequivalentstatementsthatareclausesXiaocongZHOU

Discrete

Jan. 7/

RulesofInferenceinPropositionalXiaocongXiaocongZHOUDiscreteJan. 7/Showthatthehypotheses(p∧q)∨randr→simplythepFallacy(謬論ofaffirmingtheTheproposition((p→q)∧q)→pisnotaitisfalsewhenpisfalseandqisTherearemanyincorrectargumentsthattreatthisasaTheytreattheargumentwithpremisesp→qandqandconclusionpasvalidargumentIsthefollowingargument“Ifyoudoeveryprobleminthisbook,thenyouwilllearnXiaocongZHOUDiscreteJan. 8/XiaocongZHOUDiscreteJan. 8/FallacyofdenyingtheTheproposition((p→q)∧?p)→?qisnotaitisfalsewhenpisfalseandpisIsthefollowingargument“YoudidnotlearndiscretemathematicsifyoudidnotdoeveryproblemXiaocongZHOUDiscreteJan. 9/XiaocongZHOUDiscreteJan. 9/RulesofInference fiedRulesofinference fiedToconcludethatP(c)istrue,giventhepremisewherecisaparticularmemberofelementscinweshowthat?xP(x)istruebytakinganarbitraryelementcfromtheandshowingthatP(c)istrueTheelementcthatweselectmustbeanarbitrary,andnotaspecific,ofUniversalgeneralizationisusedimplicitlyinmanyproofsinandisseldommentionedHowever,theerrorofaddingunwarrantedassumptionsabouttheelementcwhenuniversalgeneralizationisusedisalltoocommoninincorrectXiaocongZHOU

Discrete

Jan. 10/RulesofInference fiedRulesofinference fiedToconcludethatthereisanelementcin forwhichP(c)istrueweknowthat?xP(x)isWecannotselectanarbitraryvalueofchere,butratheritmustbeacforP(c)isUsuallywehavenoknowledgeofwhatcis,onlythatitBecauseitexists,wemaygiveitaname(c)andcontinueourToconcludethat?xP(x)istruewhenaparticularelementcwithP(c)trueisIfweknowoneelementcinthe forwhichP(c)istrue,thenweknowthat?xP(x)istrueXiaocongZHOU

Discrete

Jan. 10/RulesofInference fiedRulesofinference fiedShowthatthe“Everyoneinthisdiscretemathematicsclasshastakenacourse“Marlaisastudentinthisclass(Marla是這個(gè)班的學(xué)生implytheXiaocongZHOU

Discrete

Jan. 10/RulesofInference fiedRulesofinference fiedShowthatthe“Astudentinthisclasshasnotreadthe“Everyoneinth