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Announcements
•  Homework1isdueToday
•  Quiz1isWednesday
•  ReadSec:on1.8(ProofMethodsandStrategy)by
Wednesday
Networking
Platform
1 Extensible
- CSE 240 – Logic
and Discrete
Mathematics
1
ValidArguments
•  Wewillshowhowtoconstructvalidarguments
intwostages;firstforproposi:onallogicand
thenforpredicatelogic.Therulesofinference
aretheessen:albuildingblockinthe
construc:onofvalidarguments.
1.  Proposi*onalLogic
InferenceRules
2.  PredicateLogic
Inferencerulesforproposi*onallogicplusaddi*onal
inferencerulestohandlevariablesandquan*fiers.
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- CSE 240 – Logic
and Discrete
Mathematics
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ArgumentsinProposi:onalLogic
•  Aargumentinproposi:onallogicisasequenceof
proposi:ons.Allbutthefinalproposi:onarecalled
premises.Thelaststatementistheconclusion.
•  Theargumentisvalidifthepremisesimplythe
conclusion.Anargumentformisanargumentthatis
validnomaMerwhatproposi:onsaresubs:tutedintoits
proposi:onalvariables.
•  Ifthepremisesarep1 ,p2, …,pn andtheconclusionisq
then
(p1 ∧ p2 ∧ … ∧ pn)→ q isatautology. •  Inferencerulesareallargumentsinsimpleargument
formsthatwillbeusedtoconstructmorecomplex
argumentforms.
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- CSE 240 – Logic
and Discrete
Mathematics
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SocratesExample
•  Wehavethetwopremises:
–  “Allmenaremortal.”
–  “Socratesisaman.”
•  Andtheconclusion:
–  “Socratesismortal.”
•  Howdowegettheconclusionfromthepremises?
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- CSE 240 – Logic
and Discrete
Mathematics
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TheArgument
•  Wecanexpressthepremises(abovetheline)and
theconclusion(belowtheline)inpredicatelogic
asanargument:
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- CSE 240 – Logic
and Discrete
Mathematics
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ModusPonens
• Corresponding
Tautology:
• 
(p ∧ (p →q)) → q
• Example:
• Let p be “It is snowing.”
• Let q be “I will study discrete math.”
• “If it is snowing, then I will study
discrete math.”
• “It is snowing.”
• “Therefore , I will study discrete
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- CSE 240 – Logic
and Discrete
Mathematics
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ModusTollens
• Corresponding
Tautology:
• 
(¬q∧(p →q))→¬p
• Example:
• Let p be “it is snowing.”
• Let q be “I will study discrete math.”
• “If it is snowing, then I will study discrete
math.”
• “I will not study discrete math.”
• “Therefore , it is not snowing.”
Networking
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7 Extensible
- CSE 240 – Logic
and Discrete
Mathematics
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Hypothe:calSyllogism
• Corresponding Tautology:
• ((p →q) ∧ (q→r))→(p→ r)
• 
• Example:
• Let p be “it snows.”
• Let q be “I will study discrete math.”
• Let r be “I will get an A.”
• “If it snows, then I will study discrete
math.”
• “If I study discrete math, I will get an A.”
• “Therefore , If it snows, I will get an A.”
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- CSE 240 – Logic
and Discrete
Mathematics
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Disjunc:veSyllogism
• Corresponding
Tautology:
• (¬p∧(p ∨q))→q
• Example:
• Let p be “I will study discrete math.”
• Let q be “I will study English literature.”
• “I will study discrete math or I will study
English literature.”
• “I will not study discrete math.”
• “Therefore , I will study English literature.”
Networking
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- CSE 240 – Logic
and Discrete
Mathematics
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Addi:on
• Corresponding
Tautology:
• 
p →(p ∨q)
• Example:
• Let p be “I will study discrete math.”
• Let q be “I will visit Las Vegas.”
• “I will study discrete math.”
• “Therefore, I will study discrete math or
I will visit
• Las Vegas.”
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and Discrete
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Simplifica:on
• Corresponding
Tautology:
• 
(p∧q) →p
• Example:
• Let p be “I will study discrete math.”
• Let q be “I will study English
literature.”
• “I will study discrete math and
English literature”
• “Therefore, I will study discrete
math.”
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Mathematics
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Conjunc:on
• Corresponding
Tautology:
•  ((p) ∧ (q)) →(p ∧ q)
• Example:
• Let p be “I will study discrete math.”
• Let q be “I will study English literature.”
• “I will study discrete math.”
• “I will study English literature.”
• “Therefore, I will study discrete math and I
will study English literature.”
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ProofTechniques–directproofs
Here’swhatyouknow:
PatisamathmajororaCSmajor.
IfPatdoesnotlikediscretemath,PatisnotaCSmajor.
IfPatlikesdiscretemath,Patissmart.
Patisnotamathmajor.
CanyouconcludePatissmart?
M∨C
((M∨C)∧(¬D→¬C)∧(D→S)∧(¬M))→S?
Extensible
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¬D→¬C
D→S
¬M
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ProofTechniques-directproofs
Ingeneral,toprovep→q,assumepand
showthatqfollows.
((M∨C)∧(¬D→¬C)∧(D→S)∧(¬M))→S?
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ProofTechniques-directproofs
1.M∨C
2.¬D→¬C
3.D→S 4.¬M 5.C
6.D
7.S
Given
Given
Given
Given
DS(1,4)
MT(2,5)
MP(3,6)
Patissmart!
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and Discrete
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ProofTechniques-directproofs
Atotallydifferentexample:
Provethatifnisodd,then5n+3iseven.
Beforeweproveit,weneedtodefineevenandodd.
Howcanwedefineanevennumber?
Theintegernisevenifthereexistsaninteger
ksuchthatn=2k
Howcanwedefineanoddnumber?
Theintegernisoddifthereexistsaninteger
ksuchthatn=2k+1
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ProofTechniques-directproofs
Provethatifnisodd,then5n+3iseven.
Supposenisodd,
Thereforen=2k+1forsomeintegerk.
Therefore5n+3= 5(2k+1)+3
=
10k+5+3=10k+8
=
2(5k+4)or2(k’)
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ProofTechniques-directproofs
Example:
Provethatifnisodd,thenn2isodd.
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ProofTechniques-vacuousproofs
Ingeneral,toprovep→q,assumepandshow
thatqfollows.
Butp→qisalsoTRUEifpisFALSE.
Ex. p:xisodd
q:x+1iseven
∀x,xodd→x+1iseven
whataboutwhenxis4?
SincepisFALSE,p→qisTRUE
(butwedon’tknowathingaboutq)
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ProofTechniques-trivialproofs
Ingeneral,toprovep→q,assumepandshowthat
qfollows.
Butp→qisalsoTRUEifqisTRUE.
Suggestsprovingp→qbyprovingq.
Ex. p:ThereisaLionintheroom
q:2+2=4
SinceqisTRUE,p→qisTRUE
(thetruthorfalsityofpisirrelevant)
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Indirectproofs–Proofbycontraposi:on
Recallthatp→q≡¬q→¬p(thecontraposi:ve)
So,wecanprovetheimplica:onp→qbyfirst
assuming¬q,andshowingthat¬pfollows.
Example:Giventhataandbareintegers,
Prove:ifa+b≥15,thena≥8orb≥8.
(a+b≥15)→(a≥8)v(b≥8)
(Assume¬q)Suppose(a<8)∧(b<8).
(Show¬p)
Then(a≤7)∧(b≤7),
and(a+b)≤14,
and(a+b)<15.(¬p)
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Mathematics
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ProofTechniques-proofbycontradic:on
Toproveaproposi:onp,assumenotpandshowa
contradic:on.
(Provethattheskyisblue…Assumethatthesky
isnotblue)
Supposetheproposi:onisoftheforma→b,and
recallthata→b≡¬avb≡¬(a∧¬b).So
assumingtheoppositeistoassumea∧¬b.
•  Foracondi:onal,weassumeaandprove¬b
•  IfIstudyhard,thenIwillearnanA
–  AssumeIstudyhardandIwillNotearnanA
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ProofTechniques-proofbycontradic:on
Example:
Rainydaysmakegardensgrow.
Gardensdon’tgrowifitisnothot.
Whenitiscoldoutside,itrains.
Provethatit’s(always)hot.
Given: R→G
¬H→¬G
¬H→R
Show:H
((R→G)∧(¬H→¬G)∧(¬H→R))→H?
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ProofTechniques-proofbycontradic:on
Given:R→G
¬H→¬G
¬H→R
Show:H
1.R→G
Given
2.¬H→¬G Given
3.¬H→R Given
4.¬H assumetothecontrary
5.R MP(3,4)
6.G MP(1,5)
7.¬G MP(2,4)
8.G∧¬G
contradic*on
∴H
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ProofTechniques-proofbycontradic:on
Classicproofthat√2isirra:onal
–  Irra*onalnumbersarethosethatcannotberepresentedasasimplefrac*on
Suppose√2isra:onal.Then√2=a/bforsomeintegersaandb(rela:velyprime)
Defini:on:aandbarerela:velyprimeiftheyhavenocommonfactorotherthan1
√2=a/bimplies
2=a2/b2
2b2=a2
Butifaandbareboth
even,thentheyarenot
rela*velyprime!
Contradic*on!
a2iseven,andsoaiseven(a=2kforsomek)
2b2=(2k)2=4k2
b2=2k2
b2iseven,andsobiseven(b=2mforsomem)
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and Discrete
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ProofTechniques–switchingbacktocontraposi:on
Iclaimedthatifa2iseven,thenaiseven,too.
Tobecomplete,weshouldprovethat,too.
Remember,toshowp→qbycontraposi*onshow¬q→¬p
Showthatifaisodd,thena2isodd
Thena=2k+1forsomeintegerk
Thena2=(2k+1)(2k+1)=4k2+4k+1=2(j)+1
forsomeintegerjanda2isodd
Thereforeifa2iseven,thenaiseven
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Sameideawithproofbycontradic:on
Iclaimedthatifa2iseven,thenaiseven,too.
Tobecomplete,weshouldprovethat,too.
Remember,toshowp→qbycontradic*onassumepand¬qtobetrue
Supposetothecontrarya2iseven,butaisnot
Thena=2k+1forsomeintegerk
Thena2=(2k+1)(2k+1)=4k2+4k+1=2(j)+1
forsomeintegerjanda2isodd
Butweknowthata2iseven.
contradic*on
Soareallyiseven.
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GroupProblem
•  Therearetwentycoinssiingonthetable,tenare
currentlyheadsandtensarecurrentlytails.Youare
siingatthetablewithablindfoldandgloveson.
Youareabletofeelwherethecoinsare,butare
unabletoseeorfeeliftheyheadsortails.
•  Youmustcreatetwosetsofcoins.Eachsetmust
havethesamenumberofheadsandtailsasthe
othergroup.Youcanonlymoveorflipthecoins,you
areunabletodeterminetheircurrentstate.
•  Howdoyoucreatetwoevengroupsofcoinswiththe
samenumberofheadsandtailsineachgroup?
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