Outline - Brown University Review.pdf– Strings and Languages (not covered previously) – Boolean...

Outline•  MathematicalNotation

–  Sets–  SequencesandTuples–  FunctionsandRelations– Graphs–  StringsandLanguages(notcoveredpreviously)–  BooleanLogic

•  ProofsandTypesofProofs–  Construction–  Contradiction–  Induction

1

FromSipserChapter0

9/5/19 Theory of Computation - Fall'19

Lorenzo De Stefani

Sets•  Asetisagroupofobjects,orderdoesnotmatter– Theobjectsarecalledelementsormembers– Examples:

•  {1,3,5},{1,3,5,…},or{x|x∈Zandxmod2≠0}

– Youshouldknowtheseoperators/concepts•  Subset(A⊂BorA⊆B)•  Cardinality:Numberelementsinset(|A|)•  Intersection(∩)andUnion(∪),ComplementĀ•  VennDiagrams:canbeusedtovisualizesets

2 9/5/19 Theory of Computation - Fall'19

Lorenzo De Stefani

SetsII

•  PowerSet:Allpossiblesubsetsofaset–  IfA={0,1}thenwhatisP(A)?–  Ingeneral,whatisthecardinalityofP(B)?


Lorenzo De Stefani

SetsII

•  PowerSet:Setofallpossiblesubsetsofaset–  IfA={0,1}thenwhatisP(A)?

•  P(A)={,{0},{1},{0,1}}–  Ingeneral,whatisthecardinalityofP(B)?

•  Numberofthepossibilebinarystringswith|B|bits|P(B)|=2|B|.

4

;<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>


Lorenzo De Stefani

SequencesandTuples

•  Asequenceisalistofobjects,ordermatters– Example:(1,3,5)or(5,3,1)

•  Inthiscoursewewillusetermtupleinstead–  (1,3,5)isa3-tupleandak-tuplehaskelements


Lorenzo De Stefani

SequencesandTuplesII

•  Cartesianproduct(x)isanoperationonsetsbutyieldsasetoftuples–  Example:ifA={1,2}andB={x,y,z}

•  AxB={(1,x),(1,y),(1,z),(2,x),(2,y),(2,z)}–  IfwehaveksetsA1,A2,…,Ak,wecantaketheCartesianproductA1x A2… x Akwhichisthesetofallk-tuples(a1,a2,…,ak)whereai∈Ai

– WecantakeCartesianproductofasetwithitself•  AkrepresentsAxAxA…xAwheretherearekA’s.

–  ThesetZ2representsZxZallpairsofintegers,whichcanbewrittenas{(a,b)|a∈Zandb∈Z}


Lorenzo De Stefani

FunctionsandRelations

•  Afunctionmapsaninputtoa(single)output–  f(a)=b,fmapsatob

•  Thesetofpossibleinputsisthedomainandthesetofpossibleoutputsistherange–  f:D→R–  Example1:fortheabsfunction,ifD=Z,whatisR?–  Example2:sumfunction

•  CansayZxZ→Z

•  Functionscanbedescribedusingtables–  Example:Describef(x)=2xforD={1,2,3,4}


Lorenzo De Stefani

Relations•  Apredicateisafunctionwithrange{True,False}

–  Example:even(4)=True•  A(k-ary)relationisapredicatewhosedomainisasetofk-tuplesAxAxA…xA–  Ifk=2thenbinaryrelation(e.g.,=,<,...)

•  Relationsmayhave3keyproperties:–  reflexive,symmetric,transitive–  Abinaryrelationisanequivalencerelationifithasall3–  Try=,<,friend


Lorenzo De Stefani

Graphs

•  AgraphisasetofverticesVandedgesE– G=(V,E)andcanusethistodescribeagraph

9

A

B

C

D

V = {A, B, C, D} E = {(A,B), (A,C), (C,D), (A,D), (B,C)}


Lorenzo De Stefani

GraphsII

•  Definitions:– Thedegreeofavertexisthenumberofedgestouchingit

– Apathisasequenceofnodesconnectedbyedges– Asimplepathdoesnotrepeatnodes– Apathisacycleifitstartsandendsatsamenode– Asimplecyclerepeatsonlyfirstandlastnode– Agraphisatreeifitisconnectedandhasnosimplecycles


Lorenzo De Stefani

StringsandLanguages

•  Thisisveryimportantforthiscourse•  Analphabetisanynon-emptyfiniteset

– Membersofthealphabetarealphabetsymbols– ∑1={0,1}– ∑2={a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z}– ∑3={0,1,a,b,c}

•  Astringoveranalphabetisafinitesequenceofsymbolsfromthealphabet– 0100isastringfrom∑1andcatisastringfrom∑2


Lorenzo De Stefani

StringsandLanguagesII

•  Thelengthofastringw,|w|isitsnumberofsymbols

•  Theemptystring,ε,haslength0•  Ifwhaslengthnthenitcanbewrittenasw1w2…wn,wherewi∈∑

•  Stringscanbeconcatenated–  abisstringaconcatenatedwithstringb–  astringxcanbeconcatenatedwithitselfktimes

•  Thisiswrittenasxk

•  Alanguageisasetofstrings


Lorenzo De Stefani

BooleanLogic

•  BooleanlogicisamathematicalsystembuiltaroundTrue(or1)andFalse(or0)

•  Belowarethebooleanoperators,whichcanbedefinedbyatruthtable–  ∧(and/conjunction) 1∧1=1;else0–  ∨(inclusiveor/disjunctions)0∨0=0;else1–  ¬(not) ¬1=0and¬0=1– →(implication) 1→0=0;else1– ↔(equality) 1↔1=1;0↔0=1

•  Canproveequalityusingtruthtables–  DeMorgan’slawandDistributivelaw


Lorenzo De Stefani

Proofs

•  Proofsareabigpartofthisclass– Aproofisaconvincinglogicalargument

•  Proofsinthisclassneedtobeclear,formalbutnotexcessively

–  Thelevelofformalismofthebookisagreatguideline!

– TypesofProofs•  A⇔BmeansAifandonlyifB

–  ProveA⇒BandproveB⇒A•  Proofbycounterexample(provefalseviaanexample)•  Proofbyconstruction(mainprooftechniquewewilluse)•  Proofbycontradiction•  Proofbyinduction


Lorenzo De Stefani

Proofs:Example1•  ProveforeverygraphGsumofdegreesofallnodesiseven

–  Takeaminutetoproveitoratleastconvinceyourselfitistrue–  Thisisaproofbyinduction

•  Theirinformalreasoning:everyedgeyouaddtouchestwoverticesandincreasesthedegreeofbothoftheseby1(i.e.,youkeepadding2)

•  SeeExample0.19p18andTheorem0.21p20–  Aproofbyinductionmeansshowing1)itistrueforsomebasecaseandthen2)iftrueforanynthenitistrueforn+1

•  Sospendaminuteformulatingtheproofbyinduction•  Basecase:0edgesinGmeanssum-degrees=0,iseven•  Inductionstep:ifsum-degreesevenwithnedgesthenshowevenwithn+1edges

–  Whenyouaddanedge,itisbydefinitionbetweentwovertices(butcanbethesame).Eachvertexthenhasitsdegreeincreaseby1,or2overall

–  evennumber+2=even(wewillacceptthatfornow)


Lorenzo De Stefani

Proofs:Example2ForanytwosetsAandB,(Theorem0.20,p20)

– Weprovesetsareequalbyshowingthattheyhavethesameelements

– Whatprooftechniquetouse?Anyideas?– Proveineachdirection:

•  Firstproveforwarddirectionthenbackwarddirections–  Showifelementxisinoneofthesetsthenitisintheother

• Wewilldoinwords,butnotasinformalasitsoundssincewearereallyusingformaldefinitionsofeachoperator

16

(A [B) = A \ B<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>


Lorenzo De Stefani

Proof:Example2•  •  Forwarddirection(LHSàRHS):

–  Assumex∈–  Thenxisnotin(A∪B)[defn.ofcomplement]–  ThenxisnotinAandxisnotinB[defn.ofunion]–  SoxisinĀandxisinandhenceisinRHS

•  Backwarddirection(RHSàLHS)–  Assumex∈–  Sox∈Āandx∈ [defn.ofintersection]–  Sox∉Aandx∉B [defn.ofcomplement]–  Soxnotinunion(A∪B) [defn.ofunion]–  Soxmustbeitscomplement[defn.ofcomplement]

•  Sowearedone!17

(A [B) = A \ B<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

(A [B)<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

B<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

B<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

A \ B<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>


Lorenzo De Stefani

Proofs:Example3

•  Foreveryevennumbern>2,thereisa3-regulargraphwithnnodes(Theorem0.22,p21)–  Agraphisk-regularifeverynodehasdegreek

•  Wewilluseaproofbyconstruction– Manytheoremssaythataspecifictypeofobjectexists.Onewaytoproveitexistsisbyconstructingit.

– Maysoundweird,butthisisbyfarthemostcommonprooftechniquewewilluseinthiscourse

•  Wemaybeaskedtoshowthatsomepropertyistrue.Wemayneedtoconstructamodelwhichmakesitclearthatthispropertyistrue


Lorenzo De Stefani

Proof:Example3continued

•  Canyouconstructsuchagraphforn=4,6,8?–  Trynow.–  Ifyouseeapattern,thengeneralizeitandthatistheproof.

–  Hint:placethenodesintoacircle•  Solution:

–  Placethenodesinacircleandthenconnecteachnodetotheonesnexttoit,whichgivesusa2-regulargraph.

–  Thenconnecteachnodetotheoneoppositeitandyouaredone.Thisisguaranteedtoworkbecauseifthenumberofnodesiseven,theoppositenodewillalwaysgethitexactlyonce.

•  Thetextdescribesitmoreformally.•  Notethatifitwasodd,thiswouldnotwork.


Lorenzo De Stefani

Proof:Example4

JackseesJill,whohascomeinfromoutside.SinceJillisnotwetheconcludesitisnotraining(Ex0.23,p22)

–  Thisisaproofbycontradiction.•  Toproveatheoremtruebycontradiction,assumeitisfalseandshowthatleadstoacontradiction

•  Inthiscase,thattranslatestoassumeitisrainingandlookforcontradiction

–  IfweknowthatifitwererainingthenJillwouldbewet,wehaveacontradictionbecauseJillisnotwet.

–  Thatistheprocess,althoughnotaverygoodexample(whatifshelefttheumbrellaatthedoor!)

–  Thiscaseisperhapsabitconfusing.Letsgotoamoremathematicalexample…


Lorenzo De Stefani

ProveSquareRootof2Irrational

•  Proofbycontradiction,assumeitisrational–  Rationalnumberscanbewrittenasm/nforintegerm,n–  Assumewithnolossofgeneralitywereducethefraction

•  Thismeansthatmandncannotbothbeeven–  Ifso,2goesintobothsoreduceit

–  Thendosomemath

•  n=m•  2n2=m2•  Thismeansthatm2isevenandthusmmustbeeven

–  Sinceoddxoddisodd

21

p2 = m/n

<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

np2 = m



Lorenzo De Stefani

ProveSquareRootof2Irrational

•  So2n2=m2andmiseven•  Anyoddnumbercanbewrittenas2kforsomeintegerk,so:

–  2n2=(2k)2=4k2Thendividebothsidesby2–  n2=2k2–  Butnowwecansaythatn2isevenandhencenmustbeeven

•  Wejustshowedthatmandnmustbothbeeven,butsincewestartedwithareducedfraction,thatisacontradiction.–  Thusitcannotbetruethat√�2 isrational

22

p2



Lorenzo De Stefani

AnotherProofbyInductionExample

•  Provethatn2≥2nforalln2,3,…•  Basecase(n=2):22≥2x2?Yes.•  Assumetrueforn=mandthenshowitmustalsobetrueforn=m+1–  Sowestartwithm2≥2mandassumeitistrue–  wemustshowthatthisrequires(m+1)2≥2(m+1)

•  Rewritingweget:m2+2m+1≥2m+2•  Simplifyingabitweget:m2≥1.•  So,weneedtoshowthatm2≥1giventhatm2≥2m

–  If2m≥1,thenwearedone.Isit?–  Yes,sincemitself≥2


Lorenzo De Stefani

Date post:	15-Oct-2020
Category:	Documents
Upload:	others
View:	0 times
Download:	0 times

Outline - Brown University Review.pdf– Strings and Languages (not covered previously) – Boolean...

Documents