CSE 240 Logic and Discrete Mathematicsbaruah/... · Extensible Networking Platform 9 9 - CSE 240...
Transcript of CSE 240 Logic and Discrete Mathematicsbaruah/... · Extensible Networking Platform 9 9 - CSE 240...
Extensible Networking Platform 1 1 - CSE 240 – Logic and Discrete Mathematics
CSE240LogicandDiscreteMathematics
Instructor:ToddSproull
DepartmentofComputerScienceandEngineeringWashingtonUniversityinSt.Louis
Extensible Networking Platform 2 2 - CSE 240 – Logic and Discrete Mathematics
Implications
• Denotedbythesymbol→– p→qcorrespondsto“Ifp,thenq”or“pimpliesq”or“qwheneverp”
• Example– IfIworkhardinthisclass,thenIwillearnanAinCSE240– IftodayisFriday,then2+3=5(TrueorFalse?Why?)– IftodayisFriday,then2+3=6(TrueorFalse?Why)
• TruthTableforimplications
p q p→ q
TTFF
TFTF
TFTT
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Truthtableforpq
p q p→ q
TTFF
TFTF
TFTT
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Moreaboutp→ q
• DifferentEnglishrepresentationsof→ pimpliesqqwheneverp
• xisodd→x+1iseven– Ifxisodd,itistrue– IfxisNOTodd,thewholestatementistrue
• PropositionscanonlybeTrueorFalse– Noundefined
p q p→ q
TTFF
TFTF
TFTT
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LogicalEquivalence
• Denotedbythesymbolp¬¬pcorrespondsto“ pislogicallyequivalenttoNOTNOTp“
• Consider¬pvq?p→q– Aretheylogicallyequivalent?– Howcanweprovethat?
p q ¬p ¬p v q p→ q
TTFF
TFTF
FFTT
TFTT
TFTT
≡≡
p q ¬p ¬p v q p→ q
TTFF
TFTF
p q ¬p ¬p v q p→ q
TTFF
TFTF
FFTT
p q ¬p ¬p v q p→ q
TTFF
TFTF
FFTT
TFTT
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Converse,Contrapositive,andInverse
• Formedasvariantsoftheconditionalstatementp→q
• Converse– q→p
• Contrapositive– ¬q→¬p
• Inverse– ¬p→¬q
• Examples
– Considertheconditionalstatement• “TheSt.LouisCardinalswinwheneveritisraining”
– qwheneverp• “Ifitisraining,thentheCardinalswin”
– ifp,thenq
– Converse• IftheCardinalswin,thenitisraining
– Contrapositive• IftheCardinalsdonotwin,thenitisnotraining
– Inverse• Ifitisnotraining,thentheCardinalsdonotwin
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BiconditionalStatement
• Denotedbythesymbol• pqcorrespondsto“pifandonlyifq”
– or“pisnecessaryandsufficientforq”– or“piffq”
• pqistrueonlywhenpandqhavethesametruthvalues
• Example– Youcantaketheflightifandonlyifyoubuyaticket
• TruthTableforbiconditionalstatement
↔↔
p q p q
TTFF
TFTF
TFFT
↔
↔
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CompoundPropositions
• Weareabletocombinemultiplepropositionstogethertobuildmorecomplicatedpropositions
• Constructatruthtableforthefollowingproposition– (p∧ q) → (¬p∨q)
p q ¬p p∧ q ¬p∨q (p ∧ q) → (¬p∨q)
TTFF
TFTF
T
T F
T T
T T
T T T T
F F F
F
F
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LogicandBitOperations
• Computersrepresentinformationusingbits– Abitisasymbolwithvalues0and1
• BooleanVariablesusevaluesofTrueorFalse– Booleanvariablescanberepresentedwithabit
• LogicaloperationscanbeperformedbyreplacingTandFwith1and0– Substitute∧ ,∨, and with AND, OR, and XOR – Example0OR1=1
• BitStringsareasequenceof0ormorebits– lengthisstringisthenumberofbits
• Abletoperformbitwiseoperationsonbitstrings
• Example111000101001111001BitwiseORBitwiseANDBitwiseXOR
TruthValue
Bit
TF
10
⊕
111 11 1101
110 11 1100001 00 0001
Ineedsomehelpwithmymath
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PropositionalLogic
• Proofsinvolvesteppingthroughamathematicalargument
• PropositionalLogicprovidessuchsteps
• Todaywewilldiscusstheprocessofmovingfromonepropositiontothenexttoformamathematicalargument
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• Challenge:Trytofindapropositionthatisequivalenttop→ q,butthatusesonlytheconnectives¬,∧,and∨
p q p→ q
TTFF
TFTF
TFTT
p q ¬p ¬pvq
TTFF
TFTF
FFTT
TFTT
LogicalEquivalence
• p→ qislogicallyequivalentto¬p∨q• orp→ q≡¬p∨ q
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• Contrapositives:p→qand¬q→¬pEx. “Ifitisnoon,thenIamhungry.” “IfIamnothungry,thenitisnotnoon.”
• Converses:p→qandq→p
Ex.“Ifitisnoon,thenIamhungry.” “IfIamhungry,thenitisnoon.”
• Inverses:p→qand¬p→¬q
Ex.“Ifitisnoon,thenIamhungry.” “Ifitisnotnoon,thenIamnothungry.”
Aretheseequivalent?
Let’stakeavote
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• Contrapositives:p→q≡¬q→¬p?Ex. “Ifitisnoon,thenIamhungry.” “IfIamnothungry,thenitisnotnoon.”
• Converses:p→q≡q→p?
Ex.“Ifitisnoon,thenIamhungry.” “IfIamhungry,thenitisnoon.
• Inverses:p→q≡¬p→¬q?
Ex.“Ifitisnoon,thenIamhungry.” “Ifitisnotnoon,thenIamnothungry.”
YES
NO
NO
Aretheseequivalent?
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Atautologyisapropositionthat’salwaysTRUE. Acontradictionisapropositionthat’salwaysFALSE.
p ¬p p ∨ ¬p p ∧ ¬p
T F F T
T T
F F
Definitions
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• Identity• Domination
• Idempotent
p ∧ T ≡ pp ∨ F ≡ p
p ∨ T ≡ Tp ∧ F ≡ F
p ∨ p ≡ pp ∧ p ≡ p
p T p ∧ T
T T T F T F
LogicalEquivalences
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• ExcludedMiddle
• Uniqueness• Doublenegation
p ∨ ¬p ≡ T
¬(¬p) ≡ p
p ∧ ¬p ≡ F
LogicalEquivalencesContinued
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• Commutativity
• Associativity• Distributivity
p ∨ q ≡
(p ∨ q) ∨ r ≡
p ∧ q ≡q ∨ p q ∧ p
(p ∧ q) ∧ r ≡p ∨ (q ∨ r) p ∧ (q ∧ r)
p ∨ (q ∧ r) ≡ p ∧ (q ∨ r) ≡
(p ∨ q) ∧ (p ∨ r) (p ∧ q) ∨ (p ∧ r)
LogicalEquivalencesContinued
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p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r)
p q r q ∧ r p ∨ (q ∧ r) p ∨ q p ∨ r (p ∨ q) ∧ (p ∨ r)
T T T T T T T T
T T F F T T T T
T F T F T T T T
T F F F T T T T
F T T T T T T T
F T F F F T F F
F F T F F F T F
F F F F F F F F
ProofofDistributivity
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• DeMorgan’sI
• DeMorgan’sII
¬(p ∨ q) ≡ ¬p ∧ ¬q
¬(p ∧ q) ≡ ¬p ∨ ¬q
DeMorgan’s
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ExampleofDeMorgan’s
p q
¬(p ∨ q) ≡ ¬p ∧ ¬q¬(p ∧ q) ≡ ¬p ∨ ¬q
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• DeMorgan’sII
¬(p ∧ q) ≡ ¬p ∨ ¬q
¬(p ∧ q) ≡ ¬(¬¬p ∧ ¬¬q) Doublenegation
≡ ¬¬(¬p ∨ ¬q) DeMorgan’s I
≡ (¬p ∨ ¬q) Doublenegation
DeMorgan’sContinued
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¬(p ∧ ¬q) ∨ q ≡ ¬p ∨ q
if NOT (blue AND NOT red) OR red then…
¬(p ∧ ¬q) ∨ q (¬p ∨ ¬¬q) ∨ q
(¬p ∨ q) ∨ q
¬p ∨ (q ∨ q)
¬p ∨ q
DeMorgan’sII
Doublenegation
Associativity
Idempotent
Proofequivalence
≡
≡
≡
≡
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Showthat[p∧(p→q)]→qisatautologyWeuse≡toshowthat[p∧(p→q)]→q≡T
substitution for → [p ∧ (p → q)] → q
distributiveuniquenessidentitysubstitution for →De Morgan’s IIassociativeexcluded middledomination
AnotherExample
≡ [p ∧ (¬p ∨ q)] → q ≡ [(p ∧ ¬p) ∨ (p ∧ q)] → q ≡ [ F ∨ (p ∧ q)] → q ≡ (p ∧ q) → q ≡ ¬(p ∧ q) ∨ q ≡ (¬p ∨ ¬q) ∨ q ≡ ¬p ∨ (¬q ∨ q )≡ ¬p ∨ T≡ T
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AnotherExample
• Considerthenewspaperheadline:“LegislatureFailstoOverrideGovernor’sVetoofBilltoCancelSalesTaxReform”
Didthelegislaturevoteinfavoroforagainstthesalestaxreform?
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AnotherExample
• Considerthenewspaperheadline:“LegislatureFailstoOverrideGovernor’sVetoofBilltoCancelSalesTaxReform”
Didthelegislaturevoteinfavoroforagainstthesalestaxreform?
A)ThelegislatureDIDvoteinfavorB)ThelegislatureDIDNOTvoteinfavor
Let’stakeavote
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AnotherExample• Considerthenewspaperheadline:“LegislatureFailstoOverrideGovernor’sVetoofBilltoCancelSalesTaxReform”Didthelegislaturevoteinfavoroforagainstthesalestaxreform?Letsstandfor“salestaxreform”Unraveloneatatime.
-Thebilltocancelsalestaxreformis¬s-Thegovernorsvetoofthebillis¬¬s-Overridingthismeans¬¬¬s-Thedoublenegationcancelout,leavingjust¬sThereforethelegislaturedoesnotsupportsalestaxreform.Hevotedinfavorof
thebill(tocancelit).
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GroupExercise
• Breakupintogroupsof2or3people
• Solvethefollowingproblem:
• ProveDeMorgan’sILawbymanipulatingsymbols(notatruthtable)
• Hint-SimilartotheproofofDeMorgan’sIILaw
• Reminder– DeMorgan’sII
¬(p ∨ q) ≡ ¬p ∧ ¬q
¬(p ∧ q) ≡ ¬p ∨ ¬q