Chapter , Part I: Propositional...
Transcript of Chapter , Part I: Propositional...
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Chapter1,PartI:PropositionalLogic
WithQuestion/AnswerAnimations
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ChapterSummary PropositionalLogic
TheLanguageofPropositions Applications LogicalEquivalences
PredicateLogic TheLanguageofQuantifiers LogicalEquivalences NestedQuantifiers
Proofs RulesofInference ProofMethods ProofStrategy
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Proposi1onalLogicSummary TheLanguageofPropositions
Connectives TruthValues TruthTables
Applications TranslatingEnglishSentences SystemSpecifications LogicPuzzles LogicCircuits
LogicalEquivalences ImportantEquivalences ShowingEquivalence Satisfiability
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Section1.1
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Sec1onSummary Propositions Connectives
Negation Conjunction Disjunction Implication;contrapositive,inverse,converse Biconditional
TruthTables
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Proposi1ons Apropositionisadeclarativesentencethatiseithertrueor
false. Examplesofpropositions:
a) TheMoonismadeofgreencheese.b) TrentonisthecapitalofNewJersey.c) TorontoisthecapitalofCanada.d) 1+0=1 e) 0+0=2
Examplesthatarenotpropositions.a) Sitdown!b) Whattimeisit?c) x+1=2d) x+y=z
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Proposi1onalLogic ConstructingPropositions
PropositionalVariables:p,q,r,s,… ThepropositionthatisalwaystrueisdenotedbyTandthepropositionthatisalwaysfalseisdenotedbyF.
CompoundPropositions;constructedfromlogicalconnectivesandotherpropositions Negation¬ Conjunction∧ Disjunction∨ Implication→ Biconditional↔
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CompoundProposi1ons:Nega1on Thenegationofapropositionpisdenotedby¬pandhasthistruthtable:
Example:Ifpdenotes“Theearthisround.”,then¬pdenotes“Itisnotthecasethattheearthisround,”ormoresimply“Theearthisnotround.”
p ¬pT F
F T
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Conjunc1on Theconjunctionofpropositionspandqisdenotedbyp ∧ q andhasthistruthtable:
Example:Ifpdenotes“Iamathome.”andqdenotes“Itisraining.”thenp ∧ qdenotes“Iamathomeanditisraining.”
p q p ∧ q T T T
T F F
F T F
F F F
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Disjunc1on Thedisjunctionofpropositionspandqisdenotedbyp ∨qandhasthistruthtable:
Example:Ifpdenotes“Iamathome.”andqdenotes“Itisraining.”thenp ∨ qdenotes“Iamathomeoritisraining.”
p q p ∨qT T T
T F T
F T T
F F F
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TheConnec1veOrinEnglish InEnglish“or”hastwodistinctmeanings.
“InclusiveOr”‐Inthesentence“StudentswhohavetakenCS202 orMath120maytakethisclass,”weassumethatstudentsneedtohavetakenoneoftheprerequisites,butmayhavetakenboth.Thisisthemeaningofdisjunction. For p ∨q to be true, either one or both of p and q must be true.
“ExclusiveOr”‐Whenreadingthesentence“Souporsaladcomeswiththisentrée,”wedonotexpecttobeabletogetbothsoupandsalad.ThisisthemeaningofExclusiveOr(Xor).Inp ⊕ q , oneofpandqmustbetrue, but not both. The truth table for ⊕ is:
p q p ⊕ qT T F
T F T
F T T
F F F
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Implica1on Ifpandqarepropositions,thenp →qisaconditionalstatementor
implicationwhichisreadas“ifp,thenq”andhasthistruthtable:
Example:Ifpdenotes“Iamathome.”andqdenotes“Itisraining.”thenp →qdenotes“IfIamathomethenitisraining.”
Inp →q,pisthehypothesis(antecedentorpremise)andqistheconclusion(orconsequence).
p q p →qT T T
T F F
F T T
F F T
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UnderstandingImplica1on Inp →q theredoesnotneedtobeanyconnectionbetweentheantecedentortheconsequent. The “meaning” of p →q dependsonlyonthetruthvaluesofpandq.
Theseimplicationsareperfectlyfine,butwouldnotbeusedinordinaryEnglish.
“Ifthemoonismadeofgreencheese,thenIhavemoremoneythanBillGates.”
“IfthemoonismadeofgreencheesethenI’monwelfare.” “If1+1=3,thenyourgrandmawearscombatboots.”
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UnderstandingImplica1on(cont) Onewaytoviewthelogicalconditionalistothinkofanobligationorcontract. “IfIamelected,thenIwilllowertaxes.” “Ifyouget100%onthefinal,thenyouwillgetanA.”
Ifthepoliticianiselectedanddoesnotlowertaxes,thenthevoterscansaythatheorshehasbrokenthecampaignpledge.Somethingsimilarholdsfortheprofessor.Thiscorrespondstothecasewherepistrueandqisfalse.
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DifferentWaysofExpressingp →qifp,thenqpimpliesqifp,qponlyifq qunless¬pqwhenpqifp qwhenp qwheneverp pissufficientforqqfollowsfrompqisnecessaryforp
anecessaryconditionforpisqasufficientconditionforqisp
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Converse,Contraposi1ve,andInverse Fromp →qwecanformnewconditionalstatements.
q →pistheconverseofp →q ¬q → ¬ pisthecontrapositiveofp →q ¬ p → ¬ qistheinverseofp →q
Example:Findtheconverse,inverse,andcontrapositiveof“Itrainingisasufficientconditionformynotgoingtotown.”
Solution:converse:IfIdonotgototown,thenitisraining.inverse:Ifitisnotraining,thenIwillgototown.contrapositive:IfIgototown,thenitisnotraining.
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Bicondi1onal Ifpandqarepropositions,thenwecanformthebiconditional
propositionp ↔ q,readas“pifandonlyifq.”Thebiconditionalp ↔ qdenotesthepropositionwiththistruthtable:
Ifpdenotes“Iamathome.”andqdenotes“Itisraining.”thenp ↔ qdenotes“Iamathomeifandonlyifitisraining.”
p q p ↔ qT T T
T F F
F T F
F F T
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ExpressingtheBicondi1onal Somealternativeways“pifandonlyifq”isexpressedinEnglish:
pisnecessaryandsufficientforq ifpthenq,andconversely piffq
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TruthTablesForCompoundProposi1ons Constructionofatruthtable: Rows
Needarowforeverypossiblecombinationofvaluesfortheatomicpropositions.
Columns Needacolumnforthecompoundproposition(usuallyatfarright)
Needacolumnforthetruthvalueofeachexpressionthatoccursinthecompoundpropositionasitisbuiltup. Thisincludestheatomicpropositions
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ExampleTruthTable Constructatruthtablefor
p q r ¬r p∨ q p∨ q → ¬r
T T T F T F
T T F T T T
T F T F T F
T F F T T T
F T T F T F
F T F T T T
F F T F F T
F F F T F T
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EquivalentProposi1ons Twopropositionsareequivalentiftheyalwayshavethesametruthvalue.
Example:Showusingatruthtablethatthebiconditionalisequivalenttothecontrapositive.
Solution:
p q ¬ p ¬ q p →q ¬q → ¬ pT T F F T T
T F F T F F
F T T F T T
F F T T F T
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UsingaTruthTabletoShowNon‐EquivalenceExample:Showusingtruthtablesthatneithertheconversenorinverseofanimplicationarenotequivalenttotheimplication.
Solution:p q ¬ p ¬ q p → q ¬ p → ¬ q q → p
T T F F T T T
T F F T F T T
F T T F T F F
F F T T F T T
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Problem Howmanyrowsarethereinatruthtablewithnpropositionalvariables?
Solution:2n We will see how to do this in Chapter 6.
Notethatthismeansthatwithnpropositionalvariables,wecanconstruct2n distinct (i.e., not equivalent) propositions.
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PrecedenceofLogicalOperatorsOperator Precedence
¬ 1
∧∨
23
→↔
45
p ∨ q → ¬r isequivalentto(p ∨ q) → ¬r Iftheintendedmeaningisp ∨(q → ¬r ) thenparenthesesmustbeused.
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Section1.2
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Applica1onsofProposi1onalLogic:Summary TranslatingEnglishtoPropositionalLogic SystemSpecifications BooleanSearching LogicPuzzles LogicCircuits
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Transla1ngEnglishSentences StepstoconvertanEnglishsentencetoastatementinpropositionallogic Identifyatomicpropositionsandrepresentusingpropositionalvariables.
Determineappropriatelogicalconnectives “IfIgotoHarry’sortothecountry,Iwillnotgoshopping.” p:IgotoHarry’s q:Igotothecountry. r:Iwillgoshopping.
Ifporqthennotr.
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ExampleProblem:Translatethefollowingsentenceintopropositionallogic:
“YoucanaccesstheInternetfromcampusonlyifyouareacomputersciencemajororyouarenotafreshman.”
OneSolution:Leta,c,andfrepresentrespectively“Youcanaccesstheinternetfromcampus,”“Youareacomputersciencemajor,”and“Youareafreshman.”
a→ (c ∨ ¬ f)
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SystemSpecifica1ons SystemandSoftwareengineerstakerequirementsinEnglishandexpresstheminaprecisespecificationlanguagebasedonlogic.
Example:Expressinpropositionallogic:“Theautomatedreplycannotbesentwhenthefilesystemisfull”
Solution:Onepossiblesolution:Letpdenote“Theautomatedreplycanbesent”andqdenote“Thefilesystemisfull.”
q→ ¬ p
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ConsistentSystemSpecifica1onsDefinition:Alistofpropositionsisconsistentifitispossibletoassigntruthvaluestothepropositionvariablessothateachpropositionistrue.
Exercise:Arethesespecificationsconsistent? “Thediagnosticmessageisstoredinthebufferoritisretransmitted.” “Thediagnosticmessageisnotstoredinthebuffer.” “Ifthediagnosticmessageisstoredinthebuffer,thenitisretransmitted.”
Solution:Letpdenote“Thediagnosticmessageisnotstoredinthebuffer.”Letqdenote“Thediagnosticmessageisretransmitted”Thespecificationcanbewrittenas: p ∨ q,p → q, ¬p.Whenpisfalseandqistrueallthreestatementsaretrue.Sothespecificationisconsistent. Whatif“Thediagnosticmessageisnotretransmittedisadded.”Solution:Nowweareadding¬qandthereisnosatisfyingassignment.So
thespecificationisnotconsistent.
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LogicPuzzles Anislandhastwokindsofinhabitants,knights,whoalwaystellthe
truth,andknaves,whoalwayslie. YougototheislandandmeetAandB.
Asays“Bisaknight.” Bsays“Thetwoofusareofoppositetypes.”
Example:WhatarethetypesofAandB?Solution:LetpandqbethestatementsthatAisaknightandBisa
knight,respectively.So,then¬prepresentsthepropositionthatAisaknaveand¬qthatBisaknave. IfAisaknight,thenpistrue.Sinceknightstellthetruth,qmustalsobe
true.Then(p ∧¬q)∨ (¬ p ∧q) wouldhavetobetrue,butitisnot.So,Aisnotaknightandtherefore¬pmustbetrue.
IfAisaknave,thenBmustnotbeaknightsinceknavesalwayslie.So,thenboth¬pand¬qholdsincebothareknaves.
RaymondSmullyan(Born1919)
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LogicCircuits(StudiedindepthinChapter12) Electroniccircuits;eachinput/outputsignalcanbeviewedasa0or1.
0representsFalse 1representsTrue
Complicatedcircuitsareconstructedfromthreebasiccircuitscalledgates.
Theinverter(NOTgate)takesaninputbitandproducesthenegationofthatbit. TheORgatetakestwoinputbitsandproducesthevalueequivalenttothedisjunctionofthetwo
bits. TheANDgatetakestwoinputbitsandproducesthevalueequivalenttotheconjunctionofthe
twobits. Morecomplicateddigitalcircuitscanbeconstructedbycombiningthesebasiccircuits
toproducethedesiredoutputgiventheinputsignalsbybuildingacircuitforeachpieceoftheoutputexpressionandthencombiningthem.Forexample:
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Section1.3
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Sec1onSummary Tautologies,Contradictions,andContingencies. LogicalEquivalence
ImportantLogicalEquivalences ShowingLogicalEquivalence
NormalForms(optional,coveredinexercisesintext) ConjunctiveNormalForm
PropositionalSatisfiability SudokuExample
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Tautologies,Contradic1ons,andCon1ngencies Atautologyisapropositionwhichisalwaystrue.
Example:p∨ ¬p Acontradictionisapropositionwhichisalwaysfalse.
Example:p∧ ¬p Acontingencyisapropositionwhichisneitheratautologynoracontradiction,suchasp
P ¬p p∨ ¬p p∧ ¬pT F T F
F T T F
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LogicallyEquivalent Twocompoundpropositionspandqarelogicallyequivalentif
p ↔ qisatautology. Wewritethisasp⇔ qorasp ≡ qwherepandqarecompound
propositions. Twocompoundpropositionspandqareequivalentifandonlyif
thecolumnsinatruthtablegivingtheirtruthvaluesagree. Thistruthtableshow¬p ∨ q isequivalenttop → q.
p q ¬p ¬p ∨ q P → qT T F T T
T F F F F
F T T T T
F F T T T
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DeMorgan’sLaws
p q ¬p ¬q (p ∨ q) ¬(p ∨ q) ¬p ∧ ¬q
T T F F T F F
T F F T T F F
F T T F T F F
F F T T F T T
ThistruthtableshowsthatDeMorgan’sSecondLawholds.
AugustusDeMorgan
1806‐1871
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KeyLogicalEquivalences IdentityLaws:,
DominationLaws:,
Idempotentlaws:,
DoubleNegationLaw:
NegationLaws:,
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KeyLogicalEquivalences(cont) CommutativeLaws:,
AssociativeLaws:
DistributiveLaws:
AbsorptionLaws:
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MoreLogicalEquivalences
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Construc1ngNewLogicalEquivalences Wecanshowthattwoexpressionsarelogicallyequivalentbydevelopingaseriesoflogicallyequivalentstatements.
ToprovethatweproduceaseriesofequivalencesbeginningwithAandendingwithB.
Keepinmindthatwheneveraproposition(representedbyapropositionalvariable)occursintheequivalenceslistedearlier,itmaybereplacedbyanarbitrarilycomplexcompoundproposition.
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EquivalenceProofsExample:ShowthatislogicallyequivalenttoSolution:
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EquivalenceProofsExample:Showthatisatautology.Solution:
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Conjunc1veNormalForm(op1onal)Example:PutthefollowingintoCNF:
Solution:1. Eliminateimplicationsigns:
3. Movenegationinwards;eliminatedoublenegation:
5. ConverttoCNFusingassociative/distributivelaws
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Proposi1onalSa1sfiability Acompoundpropositionissatisfiableifthereisanassignmentoftruthvaluestoitsvariablesthatmakeittrue.Whennosuchassignmentsexist,thecompoundpropositionisunsatisfiable.
Acompoundpropositionisunsatisfiableifandonlyifitsnegationisatautology.
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Ques1onsonProposi1onalSa1sfiabilityExample:Determinethesatisfiabilityofthefollowingcompoundpropositions:
Solution:Satisfiable.AssignTtop, q, andr.
Solution:Satisfiable.AssignTtop and F to q.
Solution:Notsatisfiable.Checkeachpossibleassignmentoftruthvaluestothepropositionalvariablesandnonewillmakethepropositiontrue.
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Nota1on
Neededforthenextexample.
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Sudoku ASudokupuzzleisrepresentedbya9×9gridmadeupofnine3×3subgrids,knownasblocks.Someofthe81cellsofthepuzzleareassignedoneofthenumbers1,2,…,9.
Thepuzzleissolvedbyassigningnumberstoeachblankcellsothateveryrow,columnandblockcontainseachoftheninepossiblenumbers.
Example
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EncodingasaSa1sfiabilityProblem Letp(i,j,n)denotethepropositionthatistruewhenthenumbernisinthecellintheithrowandthejthcolumn.
Thereare9×9×9=729suchpropositions. Inthesamplepuzzlep(5,1,6)istrue,butp(5,j,6)isfalseforj=2,3,…9
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Encoding(cont) Foreachcellwithagivenvalue,assertp(d,j,n),whenthecellinrowiandcolumnjhasthegivenvalue.
Assertthateveryrowcontainseverynumber.
Assertthateverycolumncontainseverynumber.
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Encoding(cont) Assertthateachofthe3x3blockscontaineverynumber.
(thisistricky‐ideasfromchapter4help) Assertthatnocellcontainsmorethanonenumber.Taketheconjunctionoverallvaluesofn,n’,i,andj,whereeachvariablerangesfrom1to9and,
of
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SolvingSa1sfiabilityProblems TosolveaSudokupuzzle,weneedtofindanassignmentoftruthvaluestothe729variablesoftheformp(i,j,n)thatmakestheconjunctionoftheassertionstrue.ThosevariablesthatareassignedTyieldasolutiontothepuzzle.
Atruthtablecanalwaysbeusedtodeterminethesatisfiabilityofacompoundproposition.Butthisistoocomplexevenformoderncomputersforlargeproblems.
Therehasbeenmuchworkondevelopingefficientmethodsforsolvingsatisfiabilityproblemsasmanypracticalproblemscanbetranslatedintosatisfiabilityproblems.