Boolean algebra

DiscoveringHistory

Discovering

Britishmathematiciansproposedin1847and1854inordertostudythelawsofthinking(logic,mathematicallogic)Mathematicalmodel.Sincethen,R.Dai

Dejinregardsitasaspecialstyle.Duetothelackofphysicalbackground,theresearchwasslow,andnewprogresswasmadeinthe1930sand1940s.Around1935,MHStonefirstpointedoutthatthereisaclearconnectionbetweenBooleanalgebraandrings,whichmakesBooleanalgebraintheoryTherehasbeenacertaindevelopment.Booleanalgebrahasapplicationsinalgebra(algebraicstructure),logicalcalculus,settheory,topologicalspacetheory,measurementtheory,probabilitytheory,functionalanalysisandotherbranchesofmathematics;after1967,ithasbeenusedintheaxiomsofoneofthebranchesofmathematicallogicItalsoplaysacertainroleinthetheoreticalresearchofchemicalsettheoryandmodeltheory.Inrecentdecades,Booleanalgebrahashadimportantapplicationsinautomationtechnology,logicdesignofelectroniccomputersandotherengineeringtechnologyfields.

In1835,20-year-oldGeorgeBullopenedaprivateschool.Inordertoprovidestudentswithnecessarymathematicscourses,hereadsometextbooksthatintroducedmathematicswithgreatinterest.Soon,hewassurprised,arethesethingsmathematics?It'sincredible.Therefore,thisyoungmanwhohadonlyreceivedpreliminarymathematicstrainingtaughthimselfthedifficult"CelestialMechanics"andtheveryabstract"AnalyticalMechanics".Becausehehasastrongsenseofthesymmetryandbeautyofalgebraicrelations,inhislonelyresearch,hefirstdiscoveredinvariantsandpublishedthisresultasapaper.Afterthepublicationofthishigh-qualitypaper,Bullstillremainedinelementaryschooltoteach,buthebegantocommunicateorcorrespondwithmanytopBritishmathematicians,includingmathematicianandlogicianDeMorgan.Morganwasinvolvedinafamouscontroversyinthefirsthalfofthe19thcentury.BullknewthatMorganwasright,soin1848hepublishedathinbooklettodefendhisfriend.Thisbookisapreviewofsomethinggreaterforhimin6years.Whenitcameout,itimmediatelyarousedMorgan'spraiseandaffirmedthatheopenedupnewandtrickyresearchsubjects.Booleanwasalreadystudyinglogicalalgebra,namelyBooleanalgebra.Hesimplifiedlogicintoanextremelyeasyandsimplealgebra.Inthiskindofalgebra,the"reasoning"ofappropriatematerialsbecomestheelementarycalculationofformulas,whicharemuchsimplerthanmostformulasusedinthesecondgradeofalgebrainmiddleschoolinthepast.Inthisway,logicitselfissubjecttomathematics.Inordertoperfecthisresearchwork,Bullmadeextraordinaryeffortsforthenext6years.In1854,hepublishedthemasterpiece"TheLawsofThinking",whenhewas39yearsold,Booleanalgebracameout,andsetanewmilestoneinthehistoryofmathematics.Likealmostallnewthings,Booleanalgebrahasnotreceivedmuchattentionafteritsinvention.FamousmathematiciansontheEuropeancontinentcontemptuouslycalleditaphilosophicallyweirdthingwithnomathematicalmeaning.TheysuspectedthatmathematiciansfromtheBritishIslescouldmakeuniquecontributionstomathematics.BulldiedshortlyafterhisJiewaspublished.Atthebeginningofthe20thcentury,Russellbelievedin"PrinciplesofMathematics"that"PuremathematicswasdiscoveredbyBooleaninaworkhecalled"TheLawofThinking"."ThisstatementimmediatelyattractedtheattentionoftheworldtoBooleanalgebra.Today,thelogicalalgebrainventedbyBooleanhasdevelopedintoamajorbranchofpuremathematics.

Indiscretemathematics,Booleanalgebra(sometimescalledBooleanlattice)isasupplementarydistributivelattice(refertothedefinitionoflattice).Youcanthinkofelementsinvariousways;themostcommonistotreatthemasAsageneralizedtruthvalue.Asasimpleexample,supposethatthreeconditionsareindependentlytrueorfalse.TheelementsofBooleanalgebracanthenspecifyexactlythosethataretrue;thenBooleanalgebraitselfwillbeacollectionofalleightpossibilities,andthewaytocombinethemtogether.

ArelatedtopicsometimescalledBooleanalgebraisBooleanlogic,whichcanbedefinedassomethingcommontoallBooleanalgebras.ItconsistsofrelationshipsthatalwaysholdbetweentheelementsofBooleanalgebra,regardlessofwhichBooleanalgebrayouhave.Becausethealgebraoflogicgatesandcertainelectroniccircuitsisalsolikethisinform,Booleanlogicisalsostudiedinengineeringandcomputerscience,justlikeinmathematicallogic.

TheoryofOperations

BasicTheory

TheoperationsonBooleanalgebraarecalledAND(and),OR(or)andNOT(not).IfthealgebraicstructureisBooleanalgebra,theseoperationsmustbehaveliketwo-elementBooleanalgebra(thetwoelementsareTRUE(true)andFALSE(false)).Alsoknownaslogicalalgebra.Boolean(Boole,G.)isamathematicaltoolproposedin1847tostudythelawsofthinking(logic).BooleanalgebrareferstothealgebrasystemB=〈B,+,·,′〉

ItcontainssetBtogetherwithtwobinaryoperations+,·andaunaryoperationdefinedonit.Booleanalgebrahasthefollowingproperties:Foranyelementa,b,cinB,thereare:

1.a+b=b+a,a·b=b·a.

2.a·(b+c)=a·b+a·c,

a+(b·c)=(a+b)·(a+c).

3.a+0=a,a·1=a.

4.a+a′=1,a·a′=0.

BooleanalgebracanalsobeabbreviatedasB=〈B,+,·,′〉.Inthecaseofnotbeingconfused,itwillalsosetBiscalledBooleanalgebra.ThesetBofBooleanalgebraBiscalledBooleanset,alsoknownasthedomainordomainofBooleanalgebra.ItisthewholeoftheobjectsstudiedbyalgebraB.Generally,Booleansetsarerequiredtohaveatleasttwodifferentelements.0And1,anditselementsareclosedtothethreeoperations+,·,′,sonotanysetcanbecomeaBooleanset.Inthecaseofafiniteset,thenumberofelementsinaBooleansetcanonlybe2n,n=0,1,2,…Binaryoperation+iscalledBooleanaddition,Booleansum,Booleanunion,Booleandisjunction,etc.;binaryoperation·calledBooleanmultiplication,Booleanproduct,Booleanintersection,Booleanconjunction,etc.;unaryoperation'iscalledBooleanComplement,Booleannegation,remainderoperationofBooleanalgebra,etc.TherearealsoothernotationsfortheoperationsymbolsofBooleanalgebra,suchas∪,∩,-;∨,∧,?etc.BecauseBooleanalgebrawithonlyoneelementhaslittlepracticalvalue,Itisusuallyassumedthat0≠1,and0iscalledthezeroelementorminimumelementofBooleanalgebra,and1istheunitelementormaximumelementofBooleanalgebra.BooleanalgebraisusuallydefinedbyHuntington'saxiomsystem,butitcanalsobedefinedbyBean'saxiomsystemorhas0Complementaryallocationwith1isdefined.

BooksonBooleanalgebra(2photos)

Example

ThesimplestBooleanalgebrahasonlytwoelements,0and1.Anddefinedbythefollowingrules(truthtable):

Itisusedinlogic,interpreting0asfalseand1Istrue,∧isand,∨isor,¬isnot.ExpressionsinvolvingvariablesandBooleanoperationsrepresenttheformofstatements.Twosuchexpressionscanbeprovedtobeequivalentusingtheaboveaxiomsifandonlyifthecorrespondingstatementformsarelogicallyequivalent.

Two-elementBooleanalgebraisalsousedincircuitdesigninelectronicengineering;here0and1representtwodifferentstatesofabitinadigitalcircuit,typicallyhighandlowvoltage.Circuitsaredescribedbyexpressionscontainingvariables.Twosuchexpressionsareequivalenttoallthevalues​​ofthesevariablesifandonlyifthecorrespondingcircuithasthesameinput-outputbehavior.Inaddition,allpossibleinput-outputbehaviorscanbemodeledusingappropriateBooleanexpressions.

Two-elementBooleanalgebraisalsoimportantinthegeneraltheoryofBooleanalgebra,becauseequationsinvolvingmultiplevariablesareuniversallytrueinallBooleanalgebras,ifandonlyifitisinthetwo-elementItistrueinBooleanalgebra(thiscanalwaysbeconfirmedbytrivialbruteforcealgorithms).Forexample,toprovethatthefollowinglaw(Consensustheorem)isuniversallyvalidinallBooleanalgebras:

(a∨b)∧(¬a∨c)∧(b∨c)≡(a∨b)∧(¬a∨c)

(a∧b)∨(¬a∧c)∨(b∧c)≡(a∧b)∨(¬a∧c)

Thepowerset(setofsubsets)ofanygivensetSformsaBooleanalgebrawithtwooperations∨:=∪(union)and∧:=∩(intersection).Thesmallestelement0istheemptysetandthelargestelement1isthesetSitself.

ThesetofallsubsetsofafiniteorcofinitesetSisaBooleanalgebra.

Foranynaturalnumbern,thesetofallpositivedivisorsofnformsadistributivelattice,ifwewritea≤bfora|b.ThislatticeisaBooleanalgebraifandonlyifnissquare-free.Thesmallestelement0ofthisBooleanalgebraisthenaturalnumber1;thelargestelement1ofthisBooleanalgebraisthenaturalnumbern.

AnotherexampleofBooleanalgebracomesfromatopologicalspace:IfXisatopologicalspace,whichisbothopenandclosed,thecollectionofallsubsetsofXhastwooperations∨:=∪(Union)and∧:=∩(intersection)Booleanalgebra.

IfRisanarbitraryring,andwedefinethesetofcentralidempotentas

A={e∈R:e2=e,ex=xe,x∈R}

ThensetAbecomesaBooleanalgebrawithtwooperationse∨f:=e+f+efande∧f:=ef.

FeaturesandExamples

TouseBooleanalgebra,youmustunderstandthefeaturesofBooleanalgebra:

• Elementsaremembersofaset.Forexample:A1,representedbyA1∈B.IfA1isnotanelementofthisset,itisrepresentedbyA1B.

• ThecompletesetisthesetX,whichcontainsalltheelementsinthesetX.

• Anemptysetmeansthatthesetdoesnotcontainanyelements.

• Asubsetisasetthatcontainssomeelements,suchasA,whichcontainstheelementsA1andA2inthesetX,representedbyA∈B.

• Unaryoperationistooperateonasingleset(suchasA),thepurposeistofindtheelementsinthecompletesetexceptforsetA,denotedby¬A.

• Thebinaryoperationistooperateontwosets(suchasA1,A2),thepurposeis:tofindthecommonelementsofthetwosets,representedbyA1∧A2;findtwoAllelementsofasetarerepresentedbyA1∨A2.

AfterunderstandingtheabstractcharacteristicsofBooleanalgebra,weuseexamplestointroducehowtouseBooleanalgebratoperformsetoperations.SetsetBcontainselements{1,2,3,4,5,6,7,8,9,10},subsetA1containselements{1,2,3,4,5},subsetA2Containselements{3,4,5,7,10}.Thenthecorrespondingoperationresultis:

• ¬B=¬1=0,whichmeansanemptyset.

• ¬A1={6,7,8,9,10}.

• A1∧A2={1,2,3,4,5}∧{3,4,5,7,10}={3,4,5}.

• A1∨A2={1,2,3,4,5}∨{3,4,5,7,10}={1,2,3,4,5,7,10}.

OrderTheory

Image:Hassediagramofpowersetof3.png

TheBooleanlatticeofthesubsetisthesameasanylattice,Booleanalgebra(A,$\backslash land$,$\backslash lor$)canleadtoapartiallyorderedset(A,≤),bydefinition

a≤bwhenAndonlyifa=a$\backslash land$b(itisalsoequivalenttob=a$\backslash lor$b).

Infact,youcanalsodefineBooleanalgebraasadistributivelattice(A,≤)withthesmallestelement0andthelargestelement1(consideredasapartiallyorderedset),inwhichallelementsxhavecomplements¬xsatisfies

x$\backslash land$¬x=0andx$\backslash lor$¬x=1

here$\backslash land$and$\backslash lor$areusedtoindicatetheinfimum(intersection)andsupremum(union)oftwoelements.Also,ifcomplementsintheabovesenseexist,theyareuniquelydeterminable.

Theviewpointsofalgebraandordertheorycanusuallybeusedinterchangeably,andbothareofimportantuse.Resultsandconceptscanbeintroducedfromuniversalalgebraandordertheory.Inmanypracticalexamples,orderrelations,conjunction(logicalAND),disjunction(logicalOR)andnegation(logicalnegation)areallnaturallyavailable,sothisconnectioncanbeuseddirectly.

Principleofduality

YoucanalsoapplythegeneralunderstandingofdualityfromordertheorytoBooleanalgebra.Inparticular,theorderdualityofallBooleanalgebras,orequivalently,thealgebrasobtainedbyswapping$\backslash land$and$\backslash lor$arealsoBooleanalgebras.Generallyspeaking,anyvalidlawofBooleanalgebracanbetransformedintoanothervalidduallawbyswapping0and1,$\backslash land$and$\backslash lor$,and≤and≥.

Othernotation

TheoperatorsofBooleanalgebracanbeexpressedinvariousways.TheyareoftensimplywrittenasAND,OR,andNOT.Whendescribingthecircuit,youcanalsouseNAND(NOTAND),NOR(NOTOR),andXOR(exclusiveOR).Mathematicians,engineers,andprogrammersoftenuse+forORand·forAND(becauseinsomerespectstheseoperationsaresimilartoadditionandmultiplicationinotheralgebraicstructures,andtheseoperationsareeasyforpeoplefamiliarwithordinaryalgebratogettheproduct.Andnormalform),andtoexpressNOTasdrawingahorizontallineontopoftheexpressiontobenegated.

Hereweuseanothercommonnotation,"cross"$\backslash land$meansAND,"and"$\backslash lor$meansOR,and¬meansNOT.(Readersusingtext-onlybrowserswillseeLaTeXcodeinsteadofthewedgenotationtheyrepresent.)

HomomorphismandIsomorphism

BetweenBooleanalgebrasAandBThehomomorphismofisafunctionf:A→B,forallaandbinA:

f(a$\backslash lor$b)=f(a)$\backslash lor$f(b)

f(a$\backslash land$b)=f(a)$\backslash land$f(b)

f(0)=0

f(1)=1

ThenforallainA,f(¬a)=¬f(a)alsoholds.AllclassesofBooleanalgebra,togetherwiththeconceptofmorphism,formacategory.TheisomorphismfromAtoBisthebijectivehomomorphismfromAtoB.Theinverseofahomomorphismisalsoahomomorphism.WecallthetwoBooleanalgebrasAandBhomomorphism.FromthestandpointofBooleanalgebratheory,theyareindistinguishable;theydifferonlyinthesignsoftheirelements.

DerivativeTheory

BooleanRing

EveryBooleanalgebra(A,$\backslash land$,$\backslash lor$)allleadtoaring(A,+,*),bydefininga+b=(a$\backslash land$¬b)$\backslash lor$(b$\backslash land$¬a)(Thisoperationiscalled"symmetricdifference"insettheoryandXOR(exclusiveOR)inlogic)anda*b=a$\backslash land$b.Thezeroelementofthisringcorrespondsto0inBooleanalgebra;themultiplicationunitelementoftheringis1inBooleanalgebra.Thisringhasthepropertyofmaintaininga*a=aforallainA;aringwiththispropertyiscalledaBooleanring.

Conversely,ifaBooleanringAisgiven,wecanconvertitintoaBooleanalgebrabydefiningx$\backslash lor$y=x+y+xyandx$\backslash land$y=xy.Becausethesetwooperationsaremutuallyinverse,wecansaythateachBooleanringinducesaBooleanalgebra,orviceversa.Inaddition,themappingf:A→BisahomomorphismofBooleanalgebraifandonlyifitisahomomorphismofaBooleanring.TheBooleanringandthecategoryofalgebraareequivalent.

TheidealofBooleanalgebraAisasubsetI.ForallxandyinI,wehavex$\backslash lor$yinI,andforallxandyinA,Forallawehavea$\backslash land$xinI.TheconceptofidealconformstotheconceptofringidealinBooleanringA.TheidealIofAiscalledprimeideal,ifI≠A;andifa$\backslash land$binIalwaysimpliesainIorbinI.TheidealIofAiscalledthemaximalideal.IfI≠AandtheonlyidealthattrulycontainsIisAitself.TheseconceptsconformtotheringtheoryconceptsofprimeidealsandmaximalidealsinBooleanringA.

Theidealdualityisafilter.ThefilterofBooleanalgebraAisthesubsetp,forallx,yinpwehavex$\backslash land$yinp,andforallainA,ifa$\backslash lor$x=athenaisinp.

Howtorepresent

ItcanbeprovedthatallfiniteBooleanalgebrasareisomorphictotheBooleanalgebrasofallsubsetsofthisfiniteset.Inaddition,thenumberofelementsinallfiniteBooleanalgebrasisapoweroftwo.

Stone’sfamousrepresentationtheoremofBooleanalgebrastatesthatallBooleanalgebrasAareisomorphictoallBooleanalgebrasofclosedopensetsinacertain(compactandcompletelydisconnectedHausdorff)topologicalspace.

Axiomatization

In1933,theAmericanmathematicianEdwardVermilyeHuntington(1874-1952)demonstratedthefollowingaxiomatizationofBooleanalgebra:

:X+y=y+x.

Associationlaw:(x+y)+z=x+(y+z).

Huntingtonequation:n(n(x)+y)+n(n(x)+n(y))=x.

Theunaryfunctionsymbolncanbepronouncedas'complement'.

HerbertRobbinsthenposesthefollowingquestion:CantheHuntingtonequationbeshortenedtothefollowingequation,andthisnewequationtogetherwiththeassociativeandcommutativelawsbecomethebasisofBooleanalgebra?TheaxiomofRobbinsalgebra,thequestionbecomes:AreallRobbinsalgebrasBooleanalgebras?

AxiomizationofRobbinsalgebras:

Commutativelaw:x+y=y+x.

Associationlaw:(x+y)+z=x+(y+z).

Robbinsequation:n(n(x+y')+n(x+n(y)))=x.

Thisquestionhasbeenpublicsincethe1930sandhasbecomethefavoritequestionofAlfredTarskiandhisstudents.

In1996,WilliamMcCuneatArgonneNationalLaboratory,builtontheworkofLarryWos,SteveWinker,andBobVeroff,definitelyansweredthislong-standingquestion:AllRobbinsalgebrasareBooleanalgebra.ThisworkisdoneusingMcCune'sautomaticreasoningprogramEQP.

Rule

SubstitutionruleItcanbedescribedasanyvariableAinalogicalalgebraicformula,whichcanbereplacedbyanotherfunctionZ,andtheequationstillholds.

ThedualityrulecanbedescribedasforanylogicalexpressionF,ifthe“+”isreplacedby“*”,“*”isreplacedby“+”,and“1”isreplacedby“0”.","0"isreplacedwith"1",andtheoriginallogicpriorityisstillmaintained,thenthedualGoftheoriginalfunctionFcanbeobtained,andFandGaredualsofeachother.Wecanseethatthebasicformulasappearinpairs,andthetwoaredual.

Theinversionrulehastheoriginalfunctionandtheinversefunctioniscalledinversion(usingMorgan’slaw).

Wecandescribetheinversionruleasfollows:"*”isreplacedby“+”,“+”isreplacedby“*”,“0”isreplacedby“1”,and“1”isreplacedby“0”;theoriginalvariableisreplacedbytheinversevariable,theinversevariableisreplacedbytheoriginalvariable,longThenon-signaturemeansthatthenon-signatureoftwoormorevariablesremainsunchanged,andtheinversefunctionoftheoriginalfunctionisobtained.

Law

Thelawofcomplementarity:

Thefirstlawofcomplementarity:IfA=0,then~A=1,ifA=1,then~A=0Note:~A=NOTA

Thesecondcomplementaritylaw:A*~A=0

Thethirdcomplementaritylaw:A+~A=1

Doublecomplementaritylaw:/=//A=A

Commutativelaw:

ANDcommutativelaw:A*B=B*A

ORcommutativelaw:A+B=B+A

AssociativeLaw:

ANDAssociativeLaw:A=C*

ORAssociativeLaw:A+=C+

DistributiveLaw:

FirstDistributiveLaw:A*=+

Thesecondlawofdistribution:A+=*

Thelawoftautology:

Thelawofthefirsttautology:A*A=AIfA=1,thenA*A=1;ifA=0,thenA*A=0.Therefore,theexpressionissimplifiedtoA

Thesecondtautology:A+A=AIfA=1,then1+1=1;ifA=0,then0+0=0.Therefore,theexpressionissimplifiedtoA

Tautologywithconstants:

A+1=1

A*1=A

A*0=0

A+0=A

Thelawofabsorption:

Thefirstlawofabsorption:A*=A

SecondAbsorptionLaw:A+=A

Prototype

ThereiskkelementsetX/i>anelementaryoperationf:X→X,soon{0,1}2nmeta-operations.SoallBooleanalgebras,regardlessofsize,aretwoconstantsor"zero-element"operations,fourunaryoperations,16binaryoperations,256ternaryoperations,andsoon.TheyarecalledtheBooleanoperation.Theonlyexceptionistheone-elementBooleanalgebra,whichiscalleddegenerateortrivial(forbiddenbysomeearlyauthors).AlloperationsofBooleanalgebracanprovetobeunique.(Inthedegeneratecase,alloperationsforagivenelementarethesameoperationsbecauseallinputsreturnthesameresult.)

Operationson{0,1}canbeextendedbytruthtablesOut,select0and1asthetruevalues​​falseandtrue.Theycanbelistedinaunifiedandapplication-independentmanner,allowingustonamethemoratleastlistthemindividually.ThesenamesprovideconvenientabbreviationsforBooleanoperations.nThenameoftheelementoperationisa2-digitbinarynumber.Therearetwosuchoperations,youcan'tgetamoreconcisenomenclature!

Thefollowingshowsthepatternandassociatednamesofalloperationswithquantitiesfrom0to2.

TruthtableofBooleanoperationsupto2yuan

Constant

Unaryoperation

Binaryoperation

Thesetablescontinuetohigheryuan,rightnTheelementhas2lines,eachlinegivesanevaluationorbindingofnvariablesx0,...xn−1,Andeachcolumnhasaheaderfi,whichgivestheithnelementoperationfi(x0,...,xn−1)Thevalueunderthisevaluation.Theoperationincludesthevariableitself,forexample,f2isx0andf10isx0(asitsunaryTwocopiesofthecounterpart)andf12isx1(thereisnoone-elementcounterpart).Negativeorcomplement¬x0appearsasf1appearsagainasf5,togetherwithf3(¬x1doesnotappearat1yuan),disjunctionorcombinationx0∨x1appearsasf14,Conjunctionorcrossingx0∧x1appearsasf8,whichimpliesx0→x1appearsasf13,XORorsymmetricdifferencex0⊕x1appearsasf6,thedifferencesetx0−x1appearsasf2andsoon.ForothernamesorrepresentationsofBooleanfunctions,pleaserefertozero-orderlogic.

Asasecondarydetailaboutitsformratherthanitscontent,analgebraicoperationistraditionallyorganizedasalist.HereweindextheoperationsofBooleanalgebrabyfiniteoperationson{0,1}.Theabove-mentionedtruthtableindicatesthesortingfirstaccordingtothenumberofyuan,andthenthelistofoperationsforeachnumberofyuan.Theorderofthelistofagivenelementnumberisdeterminedbythefollowingtworules.

• (i)Therowiinthelefthalfofthetableisthebinaryrepresentationofi,withtheleastsignificantbitor0Itisatthefarleft(the"littleendian"orderwasoriginallyproposedbyAlanTuring,soitisnotunreasonablycalledtheTuringorder).

• (ii)Isthecolumnjintherighthalfofthetablethebinaryrepresentationofj,orinlittleendianorder.Thesubscriptoftheoperationontheeffectisthetruthtableofthisoperation.

Applicationfield

Application

Booleanalgebracannotonlyimplementsetoperationsinthefieldofmathematics,Whichismorewidelyusedinlogicoperationsinthefieldsofelectronics,computerhardware,computersoftware,etc.:Whenthesetcontainsonlytwoelements(1and0),theycorrespondto{true}and{false},whichcanbeusedtoimplementlogicaloperations.Judgment.

Commonapplicationsinclude:

• Digitalcircuitdesign,0and1correspondtothestateofabitinthedigitalcircuit,forexample:highlevel,lowlevelLevel.

• Thecomputer'snetworksettingsusethebinarycharacteristicsofthecomputertoperformalogicalANDoperationbetweenthesubnetmaskandthelocalIPaddresstoobtainthecomputer'snetworkaddressandhostaddress.

• DatabaseapplicationsrequirelogicaloperationstodeterminespecificquerytargetswhenqueryingthedatabasethroughSQLstatements.