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SetsandLogic
CSC1300–DiscreteStructuresVillanovaUniversity
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MajorThemes• Sets
– Waysofdefiningsets– Subsets,complements,theuniversalset– Venndiagrams– Proofsofsetequalityviadoubleinclusion
• Logic– ProposiJons– Truthtables– Venndiagrams– QuanJfiers– Prooftechniques:direct,indirect,contradicJon
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BasicterminologyAsetisanunorderedcollecJonofdisJnctobjectscalledelementsormembersoftheset.ThecardinalityofafinitesetSisdenoted|S|.ThenotaJonx∈S—means“xisanelementofS” Example:S={2,4,6,8},|S|=42∈S—“2isanelementofS”3∉S—“3isnotanelementofS”Example:S={{2,4},{6},8},|S|=3{2,4}∈S—“{2,4}isanelementofS”2∉S—“2isnotanelementofS”Amul0setorabagisanunorderedcollecJonofobjectsthatarenotnecessarilydisJnct.
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Someimportantsets
• ℕ={1,2,3,…}-thesetofnaturalnumbers• ℤ={…,-2,-1,0,1,2,…}-thesetofintegers• 𝕎={0,1,2,3,…}-thesetofposi0veintegers• ℤ2={0,1}-thebinarydigits• ℝ-thesetofrealnumbers• ℚ={x|x=p/qwherep,q∈Z,q≠0}-thesetofra0onalnumbers
• Theempty(ornull)set,denotedby∅,or{}.VillanovaCSC1300-DrPapalaskari
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Describingsets
Twowaystodescribeaset:1. BylisJngelements,e.g.,S={2,4,6,8}
2. Byaproperty,e.g.,T={x|xisanevenposiJveinteger}
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SubsetsSisasubsetofT,denotedS⊆T,iffeveryelementofSisalsoanelementofT.Examples:{a,b}⊆{a,b,c}{a,b,c}⊆{a,b,c}
ℤ ⊆ ℚS⊆S(foreveryS)∅⊆S (foreveryS)SisapropersubsetofT,denotedS⊂T,iffSisasubsetofTbutnotviceversa.Examples:{a,b}⊂{a,b,c}{b}⊂{a,b,c}whatabout∅⊂S???NotethatS⊂TiffS⊆TandS≠T
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ThePowerSet
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ThepowersetofasetSisthesetofallsubsetsofS.ThepowersetofSisdenotedbyP(S).
P(∅)={∅}
P({a})={∅,{a}}
P({a,b})={∅,{a},{b},{a,b}}
P({0,1,2})={∅,{0},{1},{2},{0,1},{0,2},{1,2},{0,1,2}}
Notethat|P(S)|=2|S|
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Setequality
TwosetsSandTareequal,denotedS=T,ifftheyhavethesameelements,i.e.,foreveryx:ifx∈Sthenx∈Tandifx∈Tthenx∈SInotherwords:
S=TiffS⊆TandT⊆S
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Prooftechnique:doubleinclusion
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Setequality
Examples:
• {a,b}={b,a}
• {1,2,3}={x|xisanintegerand0<x<4}
• {2,4,6}={x|x=2*y,wherey∈{1,2,3}}
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TheUniversalSet
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TwosetsSandTareequal,denotedS=T,ifftheyhavethesameelements,i.e.,foreveryx:ifx∈Sthenx∈Tandifx∈Tthenx∈S
Whatdoesthisevenmean????
TwosetsSandTareequal,denotedS=T,ifftheyhavethesameelements,i.e.,foreveryx:ifx∈Sthenx∈Tandifx∈Tthenx∈S
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TheUniversalSetUWeusuallythinkofsetsassubsetsofauniversalsetU.• Example:{a,b}and{b,d,e}èU={a,b,c,d,e}
(ormaybeU={a,b,c,d,e,f,g,…,z}-usuallydeterminedbycontext)ThecomplementofS,denotedSisthesetofelementsofUthatarenotinS.Example:{b,d,e}={a,c}Thesetdifference,denotedS–T(orS\T),isthesetofelementsofSthatareNOTalsoinT.Examples:{a,b,c,d,e}–{b,d,e}={a,c}(Note:S=U–S)
{b,c}–{a,b}={c}
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Venndiagrams
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U
S T S⊆T
S TS T
disjointsetsSandTSandTarenotdisjoint
U U
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SetUnionandIntersecJon
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S∪T ={x|x∈Sorx∈T} S∩T={x|x∈Sandx∈T}
S T S T
S∪Tisshaded S∩Tisshaded
Example:LetS={1,2,3,4}andT={2,3,5}.Then
S∪T={1,2,3,4,5} S∩T= {2,3}
U U
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Setdifferenceandcomplement
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S–T ={x|x∈Sandx∉T}
S T
S =U-S
S
U U
Example:LetU=NS={x|xisanintegergreaterthan6}T={x|xisanevenposiJveinteger}
ThenS-T={x|xisanoddintegergreaterthan6}S={x|xisanintegerlessthanorequalto6}
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GeneralizedunionsandintersecJons
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S1∩S2∩...∩Sndenotedby∩Sii=1
n
S1∪S2∪...∪Sndenotedby∪Si i=1
n
i=1
n
∩Si=and∪Si=i=1
n
Example:LetSi={i}.
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Generalizedunions/intersecJons
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AnotherExample:LetAi={k|k=p / i,p∈ℤ}.
n
∩Ai =and∪Ai=i=1i=1
n
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Setsandcardinality
LetA={a,b,c},B={1,2}cardinalityofaset=numberofmembers
|A|=3|B|=2
AUB={a,b,c,1,2}AB=∅U
SumPrinciple:IfAandBaredisjoint |AUB|=|A|+|B|
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review
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Setsandcardinality
LetA={a,b,c,d,e},B={b,d}|A|=5
|B|=2A\B={a,c,e}DifferencePrinciple:IfA⊆B,
|A\B|=|A|-|B| VillanovaCSC1300-DrPapalaskari
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Cartesianproduct
LetA={a,b,c},B={1,2}Thecartesianproductisthesetoforderedpairs(x,y)wherexεAandyεB:AxB={(a,1),(a,2),(b,1),(b,2),(c,1),(c,2)}
ProductPrinciple:|AxB|=|A|·|B|
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review
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Orderedpairsandn-tuples
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orderedpairs(a1,a2)andorderedn-tuples(a1,a2,…,an)• representsequenceswheretheorderofelements
doesmamerandrepeJJonsareallowed.
TheCartesianproductofthesetsS1,S2,…,Sn,denotedbyS1×S2×…×Sn,isthesetofallorderedn-tuples(s1,s2,…,sn)wheres1∈S1,s2∈S2,…,sn∈Sn.Inotherwords,S1×S2×…×Sn={(s1,s2,…,sn)|s1∈S1ands2∈S2and…andsn∈Sn}
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SetidenJJes
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S∪∅=SIdenJtyS∩U=SlawsS∪U=UDominaJonS∩∅=∅lawsS∪S=SIdempotentS∩S=SlawsS∪T=T∪SCommutaJveS∩T=T∩Slaws
ComplementaJon(S)=Slaw
S∪(T∪R)=(S∪T)∪RAssociaJveS∩(T∩R)=(S∩T)∩RlawsS∩(T∪R)=(S∩T)∪(S∩R)DistribuJveS∪(T∩R)=(S∪T)∩(S∪R)lawsS∪T=S∩TDeMorgan’sS∩T=S∪Tlaws
|S∪T|=|S|+|T|-|S∩T|Inclusion-exclusion
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ProvingsetidenJJes-example
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ProvethatS∩T=S∪T(deMorgan’sLawforsets).Proof:Weproceedbyshowingthateachsetisasubsetoftheother,i.e.S∩T⊆S∪TandS∪T⊆S∩T
1.Supposex∈S∩T. i.e.x∉S∩T. Thenx∉Sorx∉T.
Hence,x∈Sorx∈T. Thismeansthatx∈S∪T.
Thus,S∩T⊆S∪T.
2.Nowsupposex∈S∪T. Thenx∈Sorx∈T.Hencex∉Sorx∉T, whichmeansthatx∉S∩T.Therefore,x∈S∩T. Thus,S∪T⊆S∩T.
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MajorThemes
• Sets– Waysofdefiningsets– Subsets,complements,theuniversalset– Venndiagrams– Proofsofsetequalityviadoubleinclusion
• Logic– ProposiJons– Truthtables– Venndiagrams– QuanJfiers
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WhyLogic?
Logic–ascienceofreasoning
• BasisofmathemaJcalreasoning-givesprecisemeaningtomathemaJcalstatements-isusedtodisJnguishbetweenvalidandinvalidmathemaJcalarguments
• ApplicaJonsinCS:-designofhardware-programming-arJficialintelligence-databases
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Adeclara0vestatementthatiseithertrueorfalseArethefollowingproposi0ons?• 1+2=3• todayismybirthday• NewYorkisthecapitaloftheUSA• 5-3+2• x+y>5• Areyouastudent?• Don’ttalk• Yourfeetareugly• Thissentenceisfalse
ProposiNon
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Compoundproposi3onsareformedfromsimplerproposiJonsusingconnec3ves,alsocalledlogicaloperators.
TheconnecJveswewillstudyare:• nega3onornotoperatordenoted¬• conjunc3onorandoperator∧ • disjunc3onororoperator∨• exclusiveororxoroperator⊕• implica3on→• bicondi3onal↔
CompoundProposiNonsandConnecNves
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NegaNonIfpisaproposiJon,thenthestatement
“Itisnotthecasethatp”isanotherproposiJon,calledthenega3onofp.ThenegaJonofp,denotedby¬pandread“notp”,istruewhenpisfalse,andisfalsewhenpistrue.
Example:WhatisthenegaJonof“TodayisWednesday”?
ThetruthtablefornegaJon:p¬pTF
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ConjuncNonTheproposiJon“pandq”,denotedbyp∧q,iscalledtheconjunc3onofpandq.Itistruewhenbothpandqaretrue,otherwiseitisfalse.
pqp∧qTTTFFTFF
ThetruthtableforconjuncJon:
Examples:TodayisWednesdayanditisraining.TodayisWednesdaybutitisnotraining.
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DisjuncNonTheproposiJon“porq”,denotedbyp∨q,iscalledthedisjunc3onofpandq.Itisfalsewhenbothpandqarefalse,otherwiseitistrue.
ThetruthtablefordisjuncJon:pqp∨qTTTFFTFF
Example:TodayisSundayoraholiday.
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ExclusiveOR(XOR)
TheproposiJonp⊕qiscalledtheexclusiveorofpandq.Itistruewhenexactlyoneofpandqistrue,otherwiseitisfalse.
Thetruthtableforexclusiveor:
pqp⊕qTTTFFTFF
Example:Thisdishcomeswithsouporsalad.
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ImplicaNonTheimplica3onorcondi3onalproposi3onp→qistheproposiJonthatisfalseonlywhenpistrueandqisfalse.piscalledthehypothesisandqiscalledtheconclusion.
ThetruthtableforimplicaJon:pqp→qTTTFFTFF
Readingsforp→q:• “ifpthenq”• “ponlyifq”• “qisnecessaryforp”• “pissufficientforq”• “pimpliesq”• “qifp”• “qwheneverp”
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ExamplesofImplicaNonWording
IfJohnisinL.A.,thenheisinCalifornia.
TobeinCalifornia,it issufficientforJohntobeinL.A.
TobeinLA,it isnecessaryforJohntobeinCalifornia.
YouwillgetanAifyoustudyhard.vs.YouwillgetanAonlyifyoustudyhard.
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MoreExamplesofImplicaNonwording:
Ifyouplaceyourorderby11:59pmDecember21st,thenweguaranteedeliverybyChristmas.
Placingyourorderby11:59pmDecember21stguaranteesdeliverybyChristmas.
WeguaranteedeliverybyChristmasifyouplaceyourorderby11:59pmDecember21st.
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MoreExamplesofImplicaNonwording:
Ifyouplaceyourorderby11:59pmDecember21st,thenweguaranteedeliverybyChristmas.
Placingyourorderby11:59pmDecember21stguaranteesdeliverybyChristmas.
WeguaranteedeliverybyChristmasifyouplaceyourorderby11:59pmDecember21st.
WeguaranteedeliverybyChristmasonlyifyouplaceyourorderby11:59pmDecember21st.
isthisthesametoo?
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BicondiNonal
Thebicondi3onalp↔qistheproposiJonthatistruewhenpandqhavethesametruthvalues,andisfalseotherwise.
ThetruthtableforbicondiJonal:
pqp↔qTTTFFTFF
Readingsforp↔q:• “pifandonlyifq”• “pisnecessaryandsufficientforq”• “ifp,thenq,andconversely”
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TruthtablesformorecomplexproposiNons
pqrr∨(q∧¬p)TTTTTFTFTTFFFTTFTFFFTFFF
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Tautology
pqr(¬p∧(p∨q))→qTTTTTFTFTTFFFTTFTFFFTFFF
-A(compound)proposi0onthatisalwaystrue(irrespec0veofthevaluesofitscomponents)
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LogicalEquivalence
pq¬p∨qp→qTTTFFTFF
• wewrite:¬p∨q≡p→qtoindicatethattheproposiJons¬p∨qandp→qarelogicallyequivalent
Wesaythattwoproposi3onsarelogicallyequivalentifftheyhavethesametruthtable.
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DeMorgan’sLawsforLogic
FirstDeMorgan’slawforlogic:¬(p∨q)≡(¬p)∧(¬q)
Example:Negate:“TodayisSundayoraholiday”
SecondDeMorgan’slawforlogic:¬(p∧q)≡Example:Negate:“TodayisSundayandaholiday”
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Converse,InverseandContraposiNve
• q→piscalledtheconverseofp→q• ¬p→¬qiscalledtheinverseofp→q• ¬q→¬piscalledthecontraposi3veofp→q
pqp→qq→p¬p→¬q¬q→¬pTTTFFTFF
Whichoftheabovearelogicallyequivalent?
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ProposiNonalfuncNonsInteresJngstatementsinvolvevariables.DefiniNonAproposi0onalfunc0onP(x)isafuncJonwhosevaluesareproposiJons,i.e.,it’sanassignmenttoeachelementxofthefuncJon’sdomainDcalledthedomainofdiscourseaproposiJon(atrueorfalsestatement).ExampleLetP(x)denotethestatement“xiseven”.
Domainofdiscourse?P(2)
P(3)VillanovaCSC1300-DrPapalaskari
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UniversalquanNfier
DefiniNon:universalquan0fica0onofP(x)“P(x)istrueforallvaluesofxinitsuniverseofdiscourse”
“forallxP(x)”
“foreveryxP(x)”∀xP(x)universalquan0fier
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ExamplesofuniversalquanNficaNon∀x(x+0=x)∀x(x2>x)∀xP(x)whereP(x)denotesthestatement“xlovesCS”
Let M(x) denote “x is mortal” and H(x) denote “x is a human”
Express the proposition: “every human is mortal”
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ExistenNalquanNfier
DefiniNon:existen0alquan0fica0onofP(x)“thereexistsanelementxinitsuniverseofdiscoursesuchthatP(x)istrue”
“thereisanxsuchthatP(x)”
“forsomexP(x)”
∃xP(x)existen0alquan0fier.
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ExamplesofexistenNalquanNficaNon
Trueorfalse?
∃x(x+x=x*x)
∃x(x=x+1)
∃xP(x)whereP(x)denotesthestatement“xlovesmath”
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Let Q(x) denote “x is a sophomore” Express the proposition: “there is a sophomore who loves math”
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GeneralizedDeMorganLawsofLogic
• ¬∀xP(x)≡∃x¬P(x)“NoteveryonelovesCS”≡“ThereissomeonewhodoesnotloveCS”
• ¬∃x¬P(x)≡∀x¬¬P(x)≡∀xP(x)
“ThereisnoonewhodoesnotloveCS”≡“EveryonelovesCS” VillanovaCSC1300-DrPapalaskari
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ExpressionswithseveralquanNfiers
Lettheuniverseofdiscoursebethesetofallstudents(ofVU).Let
C(x)means“xhasacomputer”F(x,y)means“xandyarefriends”
TranslatethefollowingintoEnglish:
• ∀xC(x)
• ∃x¬∃yF(x,y)
• ∀x[C(x)∨∃y(F(x,y)∧C(y))]
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DoestheorderofthequanNfiersmaber?
—No,ifwehaveseveralconsecuJvequanJfiersofthesametype:∀x∀yQ(x,y)≡∀y∀xQ(x,y)∃x∃yQ(x,y)≡∃y∃xQ(x,y)—Yes,ifwehavedifferentquanJfiers:∀x∃yQ(x,y)≡∃y∀xQ(x,y)
Counterexample:LetQ(x,y)mean“x+y=0”,andlettheuniverseofdiscoursebethesetofallrealnumbers.Whatisthetruthvalueof:
∀x∃yQ(x,y)?∃y∀xQ(x,y)? VillanovaCSC1300-DrPapalaskari
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theorems
proved to be true
MathemaJcalSystem
axioms
assumed to be true
definitions
used to create terms
Logic is a tool for the analysis
of inference
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BasicTerminology
• Axiom (postulate) – underlying assumption, does not require a proof • Rules of inference – used to draw conclusions from other assertions • Proof of a statement A – a sequence of statements, each of which is:
• an axiom or • follows from one or more earlier statements
and the last statement in the sequence is A
• Informal proof vs. formal – uses rules of inference informally and formally, respectively
• Theorem – a statement that has been proved
• Lemma – a theorem used in the proof of other theorems • Corollary – a theorem that immediately follows from another theorem
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Typesofproofsn direct n indirect (by contrapositive)
n by contradiction
n proof of equivalence
n proof by cases n proof by mathematical induction
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Provingp → q
• DirectProof
• IndirectProof/ContraposiJve
• ProofbyContradicJon
p → q
p → q ≡ ¬ q → ¬ p
p → q ≡ (p ∧ ¬ q) → (r ∧ ¬ r)
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DirectProof
Toprovep → q:Supposepistrue;provethatqmustalsobetrue
Example: If n is even, then n2 is also even
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IndirectProof
Provep → qbyprovingcontraposi0ve:¬ q → ¬ pExample: If n·m is odd, then an n×m grid cannot be tiled with
dominoes.
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ContradicJonProof
Provesbyshowingthat¬sisabsurd!• ¬s→F(Reduc0oadabsurdum)Example:
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Proofsofequivalence
A proof of equivalence of two assertions (i.e., p ↔ q), often stated by using “if and only if” or “necessary and sufficient,” requires two separate parts:
p → q and q → p. • Example: An integer n is odd iff n2 is odd.
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ProofsbycasesA proof by cases is based on partitioning the theorem’s domain into subdomains and proving the theorem separately for each of these subdomains. Definition: ⌊x⌋, called the floor of x, is the largest integer ≤ x; ⌈x⌉, called the ceiling of x, is the smallest integer ≥ x. Example:
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Proofs,examples,andcounterexamples:∀xP(x)
For universal statements:
• Checking validity of a theorem for specific examples does NOT constitute a proof (unless the examples exhaust all the values in the theorem’s domain, which is impossible if the latter is infinite).
• Just a single example suffices to disprove a theorem. (Such an example is usually called a counterexample).
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Proofs,examples,andcounterexamples∃x P(x)
For existential statements:
• A single example suffices to prove the theorem (constructive proof).
• Alternatively, using contradiction, prove that it is not possible for such a thing not to exist. (non-constructive proof) • Show that a player in a game has a winning strategy without actually saying what it is! • Famous proof: There exist irrational x, y such that xy is rational
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WhichProofMethod?
1. Beginwithadirectproofapproach2. Ifthisfails,tryeither
– indirect/contraposiJveapproach– proofbycontradicJon– proofbycases– acombinaJon...
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MajorThemes• Sets
– Waysofdefiningsets– Subsets,complements,theuniversalset– Venndiagrams– Proofsofsetequalityviadoubleinclusion
• Logic– ProposiJons– Truthtables– Venndiagrams– QuanJfiers– Prooftechniques:direct,indirect,contradicJon
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