Chapter1,PartI:PropositionalLogic

WithQuestion/AnswerAnimations

ChapterSummary PropositionalLogic

 TheLanguageofPropositions Applications  LogicalEquivalences

 PredicateLogic TheLanguageofQuantifiers  LogicalEquivalences NestedQuantifiers

 Proofs RulesofInference  ProofMethods  ProofStrategy

Proposi1onalLogicSummary  TheLanguageofPropositions

  Connectives  TruthValues  TruthTables

 Applications  TranslatingEnglishSentences  SystemSpecifications  LogicPuzzles  LogicCircuits

  LogicalEquivalences  ImportantEquivalences  ShowingEquivalence  Satisfiability

Section1.1

Sec1onSummary Propositions Connectives

 Negation Conjunction Disjunction Implication;contrapositive,inverse,converse Biconditional

 TruthTables

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

Proposi1onalLogic ConstructingPropositions

 PropositionalVariables:p,q,r,s,… ThepropositionthatisalwaystrueisdenotedbyTandthepropositionthatisalwaysfalseisdenotedbyF.

 CompoundPropositions;constructedfromlogicalconnectivesandotherpropositions  Negation¬  Conjunction∧  Disjunction∨  Implication→  Biconditional↔

CompoundProposi1ons:Nega1on Thenegationofapropositionpisdenotedby¬pandhasthistruthtable:

 Example:Ifpdenotes“Theearthisround.”,then¬pdenotes“Itisnotthecasethattheearthisround,”ormoresimply“Theearthisnotround.”

p ¬pT F

F T

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

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

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

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

UnderstandingImplica1on Inp →q theredoesnotneedtobeanyconnectionbetweentheantecedentortheconsequent. The “meaning” of p →q dependsonlyonthetruthvaluesofpandq.

 Theseimplicationsareperfectlyfine,butwouldnotbeusedinordinaryEnglish.

  “Ifthemoonismadeofgreencheese,thenIhavemoremoneythanBillGates.”

  “IfthemoonismadeofgreencheesethenI’monwelfare.”  “If1+1=3,thenyourgrandmawearscombatboots.”

UnderstandingImplica1on(cont) Onewaytoviewthelogicalconditionalistothinkofanobligationorcontract.  “IfIamelected,thenIwilllowertaxes.”  “Ifyouget100%onthefinal,thenyouwillgetanA.”

  Ifthepoliticianiselectedanddoesnotlowertaxes,thenthevoterscansaythatheorshehasbrokenthecampaignpledge.Somethingsimilarholdsfortheprofessor.Thiscorrespondstothecasewherepistrueandqisfalse.

DifferentWaysofExpressingp →qifp,thenqpimpliesqifp,qponlyifq qunless¬pqwhenpqifp qwhenp qwheneverp pissufficientforqqfollowsfrompqisnecessaryforp

anecessaryconditionforpisqasufficientconditionforqisp

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.

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

ExpressingtheBicondi1onal Somealternativeways“pifandonlyifq”isexpressedinEnglish:

  pisnecessaryandsufficientforq  ifpthenq,andconversely  piffq

TruthTablesForCompoundProposi1ons Constructionofatruthtable: Rows

  Needarowforeverypossiblecombinationofvaluesfortheatomicpropositions.

 Columns Needacolumnforthecompoundproposition(usuallyatfarright)

 Needacolumnforthetruthvalueofeachexpressionthatoccursinthecompoundpropositionasitisbuiltup.  Thisincludestheatomicpropositions

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

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

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

Problem Howmanyrowsarethereinatruthtablewithnpropositionalvariables?

Solution:2n We will see how to do this in Chapter 6.

 Notethatthismeansthatwithnpropositionalvariables,wecanconstruct2n distinct (i.e., not equivalent) propositions.

PrecedenceofLogicalOperatorsOperator Precedence

¬ 1

∧∨

23

→↔

45

p ∨ q → ¬r isequivalentto(p ∨ q) → ¬r Iftheintendedmeaningisp ∨(q → ¬r ) thenparenthesesmustbeused.

Section1.2

Applica1onsofProposi1onalLogic:Summary TranslatingEnglishtoPropositionalLogic SystemSpecifications BooleanSearching LogicPuzzles LogicCircuits

Transla1ngEnglishSentences StepstoconvertanEnglishsentencetoastatementinpropositionallogic Identifyatomicpropositionsandrepresentusingpropositionalvariables.

 Determineappropriatelogicalconnectives “IfIgotoHarry’sortothecountry,Iwillnotgoshopping.” p:IgotoHarry’s q:Igotothecountry. r:Iwillgoshopping.

Ifporqthennotr.

ExampleProblem:Translatethefollowingsentenceintopropositionallogic:

“YoucanaccesstheInternetfromcampusonlyifyouareacomputersciencemajororyouarenotafreshman.”

OneSolution:Leta,c,andfrepresentrespectively“Youcanaccesstheinternetfromcampus,”“Youareacomputersciencemajor,”and“Youareafreshman.”

a→ (c ∨ ¬ f)

SystemSpecifica1ons SystemandSoftwareengineerstakerequirementsinEnglishandexpresstheminaprecisespecificationlanguagebasedonlogic.

Example:Expressinpropositionallogic:“Theautomatedreplycannotbesentwhenthefilesystemisfull”

Solution:Onepossiblesolution:Letpdenote“Theautomatedreplycanbesent”andqdenote“Thefilesystemisfull.”

q→ ¬ p

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.

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)

LogicCircuits(StudiedindepthinChapter12)  Electroniccircuits;eachinput/outputsignalcanbeviewedasa0or1.

  0representsFalse  1representsTrue

  Complicatedcircuitsareconstructedfromthreebasiccircuitscalledgates.

 Theinverter(NOTgate)takesaninputbitandproducesthenegationofthatbit. TheORgatetakestwoinputbitsandproducesthevalueequivalenttothedisjunctionofthetwo

bits. TheANDgatetakestwoinputbitsandproducesthevalueequivalenttotheconjunctionofthe

twobits.  Morecomplicateddigitalcircuitscanbeconstructedbycombiningthesebasiccircuits

toproducethedesiredoutputgiventheinputsignalsbybuildingacircuitforeachpieceoftheoutputexpressionandthencombiningthem.Forexample:

Section1.3

Sec1onSummary Tautologies,Contradictions,andContingencies. LogicalEquivalence

 ImportantLogicalEquivalences ShowingLogicalEquivalence

 NormalForms(optional,coveredinexercisesintext) ConjunctiveNormalForm

 PropositionalSatisfiability SudokuExample

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

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

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

KeyLogicalEquivalences IdentityLaws:,

 DominationLaws:,

 Idempotentlaws:,

 DoubleNegationLaw:

 NegationLaws:,

KeyLogicalEquivalences(cont) CommutativeLaws:,

 AssociativeLaws:

 DistributiveLaws:

 AbsorptionLaws:

MoreLogicalEquivalences

Construc1ngNewLogicalEquivalences Wecanshowthattwoexpressionsarelogicallyequivalentbydevelopingaseriesoflogicallyequivalentstatements.

 ToprovethatweproduceaseriesofequivalencesbeginningwithAandendingwithB.

 Keepinmindthatwheneveraproposition(representedbyapropositionalvariable)occursintheequivalenceslistedearlier,itmaybereplacedbyanarbitrarilycomplexcompoundproposition.

EquivalenceProofsExample:ShowthatislogicallyequivalenttoSolution:

EquivalenceProofsExample:Showthatisatautology.Solution:

Conjunc1veNormalForm(op1onal)Example:PutthefollowingintoCNF:

Solution:1.  Eliminateimplicationsigns:

3.  Movenegationinwards;eliminatedoublenegation:

5.  ConverttoCNFusingassociative/distributivelaws

Proposi1onalSa1sfiability Acompoundpropositionissatisfiableifthereisanassignmentoftruthvaluestoitsvariablesthatmakeittrue.Whennosuchassignmentsexist,thecompoundpropositionisunsatisfiable.

 Acompoundpropositionisunsatisfiableifandonlyifitsnegationisatautology.

Ques1onsonProposi1onalSa1sfiabilityExample:Determinethesatisfiabilityofthefollowingcompoundpropositions:

Solution:Satisfiable.AssignTtop, q, andr.

Solution:Satisfiable.AssignTtop and F to q.

Solution:Notsatisfiable.Checkeachpossibleassignmentoftruthvaluestothepropositionalvariablesandnonewillmakethepropositiontrue.

Nota1on

Neededforthenextexample.

Sudoku ASudokupuzzleisrepresentedbya9×9gridmadeupofnine3×3subgrids,knownasblocks.Someofthe81cellsofthepuzzleareassignedoneofthenumbers1,2,…,9.

 Thepuzzleissolvedbyassigningnumberstoeachblankcellsothateveryrow,columnandblockcontainseachoftheninepossiblenumbers.

 Example

EncodingasaSa1sfiabilityProblem Letp(i,j,n)denotethepropositionthatistruewhenthenumbernisinthecellintheithrowandthejthcolumn.

 Thereare9×9×9=729suchpropositions. Inthesamplepuzzlep(5,1,6)istrue,butp(5,j,6)isfalseforj=2,3,…9

Encoding(cont) Foreachcellwithagivenvalue,assertp(d,j,n),whenthecellinrowiandcolumnjhasthegivenvalue.

 Assertthateveryrowcontainseverynumber.

 Assertthateverycolumncontainseverynumber.

Encoding(cont) Assertthateachofthe3x3blockscontaineverynumber.

(thisistricky‐ideasfromchapter4help) Assertthatnocellcontainsmorethanonenumber.Taketheconjunctionoverallvaluesofn,n’,i,andj,whereeachvariablerangesfrom1to9and,

of

SolvingSa1sfiabilityProblems TosolveaSudokupuzzle,weneedtofindanassignmentoftruthvaluestothe729variablesoftheformp(i,j,n)thatmakestheconjunctionoftheassertionstrue.ThosevariablesthatareassignedTyieldasolutiontothepuzzle.

 Atruthtablecanalwaysbeusedtodeterminethesatisfiabilityofacompoundproposition.Butthisistoocomplexevenformoderncomputersforlargeproblems.

 Therehasbeenmuchworkondevelopingefficientmethodsforsolvingsatisfiabilityproblemsasmanypracticalproblemscanbetranslatedintosatisfiabilityproblems.