NAND logic - Wikipedia

XNOR NANDlogic FromWikipedia,thefreeencyclopedia Jumptonavigation Jumptosearch ThisarticleisaboutNANDlogicinthesenseofbuildingotherlogicgatesusingjustNANDgates.ForNANDgates,seeNANDgate.ForNANDinthepurelylogicalsense,seeLogicalNAND.Forlogicgatesgenerally,seeLogicgate. Thisarticlerelieslargelyorentirelyonasinglesource.Relevantdiscussionmaybefoundonthetalkpage.Pleasehelpimprovethisarticlebyintroducingcitationstoadditionalsources.Findsources: "NANDlogic" – news ·newspapers ·books ·scholar ·JSTOR(July2012) TheNANDBooleanfunctionhasthepropertyoffunctionalcompleteness.ThismeansthatanyBooleanexpressioncanbere-expressedbyanequivalentexpressionutilizingonlyNANDoperations.Forexample,thefunctionNOT(x)maybeequivalentlyexpressedasNAND(x,x).Inthefieldofdigitalelectroniccircuits,thisimpliesthatitispossibletoimplementanyBooleanfunctionusingjustNANDgates. ThemathematicalproofforthiswaspublishedbyHenryM.Shefferin1913intheTransactionsoftheAmericanMathematicalSociety(Sheffer1913).AsimilarcaseappliestotheNORfunction,andthisisreferredtoasNORlogic. Contents 1NAND 2MakingothergatesbyusingNANDgates 2.1NOT 2.2AND 2.3OR 2.4NOR 2.5XOR 2.6XNOR 3MUX 4DEMUX 5Seealso 6Externallinks 7References NAND[edit] Mainarticle:NANDgate ANANDgateisaninvertedANDgate.Ithasthefollowingtruthtable: Q=ANANDB TruthTable InputA InputB OutputQ 0 0 1 0 1 1 1 0 1 1 1 0 ACMOStransistorNANDelement.Vdddenotespositivevoltage. InCMOSlogic,ifbothoftheAandBinputsarehigh,thenboththeNMOStransistors(bottomhalfofthediagram)willconduct,neitherofthePMOStransistors(tophalf)willconduct,andaconductivepathwillbeestablishedbetweentheoutputandVss(ground),bringingtheoutputlow.IfbothoftheAandBinputsarelow,thenneitheroftheNMOStransistorswillconduct,whilebothofthePMOStransistorswillconduct,establishingaconductivepathbetweentheoutputandVdd(voltagesource),bringingtheoutputhigh.IfeitheroftheAorBinputsislow,oneoftheNMOStransistorswillnotconduct,oneofthePMOStransistorswill,andaconductivepathwillbeestablishedbetweentheoutputandVdd(voltagesource),bringingtheoutputhigh.Astheonlyconfigurationofthetwoinputsthatresultsinalowoutputiswhenbotharehigh,thiscircuitimplementsaNAND(NOTAND)logicgate. MakingothergatesbyusingNANDgates[edit] ANANDgateisauniversalgate,meaningthatanyothergatecanberepresentedasacombinationofNANDgates. NOT[edit] Seealso:NOTgate ANOTgateismadebyjoiningtheinputsofaNANDgatetogether.SinceaNANDgateisequivalenttoanANDgatefollowedbyaNOTgate,joiningtheinputsofaNANDgateleavesonlytheNOTgate. DesiredNOTGate NANDConstruction Q=NOT(A) =ANANDA TruthTable InputA OutputQ 0 1 1 0 AND[edit] Seealso:ANDgate AnANDgateismadebyinvertingtheoutputofaNANDgateasshownbelow. DesiredANDGate NANDConstruction Q=AANDB =(ANANDB)NAND(ANANDB) TruthTable InputA InputB OutputQ 0 0 0 0 1 0 1 0 0 1 1 1 OR[edit] Seealso:ORgate IfthetruthtableforaNANDgateisexaminedorbyapplyingDeMorgan'sLaws,itcanbeseenthatifanyoftheinputsare0,thentheoutputwillbe1.TobeanORgate,however,theoutputmustbe1ifanyinputis1.Therefore,iftheinputsareinverted,anyhighinputwilltriggerahighoutput. DesiredORGate NANDConstruction Q=AORB =(ANANDA)NAND(BNANDB) TruthTable InputA InputB OutputQ 0 0 0 0 1 1 1 0 1 1 1 1 NOR[edit] Seealso:NORgate ANORgateisanORgatewithaninvertedoutput.OutputishighwhenneitherinputAnorinputBishigh. DesiredNORGate NANDConstruction Q=ANORB =[(ANANDA)NAND(BNANDB)]NAND[(ANANDA)NAND(BNANDB)] TruthTable InputA InputB OutputQ 0 0 1 0 1 0 1 0 0 1 1 0 XOR[edit] Seealso:XORgate AnXORgateismadebyconnectingfourNANDgatesasshownbelow.ThisconstructionentailsapropagationdelaythreetimesthatofasingleNANDgate. DesiredXORGate NANDConstruction Q=AXORB =[ANAND(ANANDB)]NAND[BNAND(ANANDB)] TruthTable InputA InputB OutputQ 0 0 0 0 1 1 1 0 1 1 1 0 Alternatively,anXORgateismadebyconsideringthedisjunctivenormalform A ⋅ B ¯ + A ¯ ⋅ B {\displaystyleA\cdot{\overline{B}}+{\overline{A}}\cdotB} ,notingfromdeMorgan'sLawthataNANDgateisaninverted-inputORgate.Thisconstructionusesfivegatesinsteadoffour. DesiredGate NANDConstruction Q=AXORB =[BNAND(ANANDA)]NAND[ANAND(BNANDB)] XNOR[edit] Seealso:XNORgate AnXNORgateismadebyconsideringthedisjunctivenormalform A ⋅ B + A ¯ ⋅ B ¯ {\displaystyleA\cdotB+{\overline{A}}\cdot{\overline{B}}} ,notingfromdeMorgan'sLawthataNANDgateisaninverted-inputORgate.ThisconstructionentailsapropagationdelaythreetimesthatofasingleNANDgateandusesfivegates. DesiredXNORGate NANDConstruction Q=AXNORB =[(ANANDA)NAND(BNANDB)]NAND(ANANDB) InputA InputB OutputQ 0 0 1 0 1 0 1 0 0 1 1 1 Alternatively,the4-gateversionoftheXORgatecanbeusedwithaninverter.Thisconstructionhasapropagationdelayfourtimes(insteadofthreetimes)thatofasingleNANDgate. DesiredGate NANDConstruction Q=AXNORB ={[ANAND(ANANDB)]NAND[BNAND(ANANDB)]}NAND{[ANAND(ANANDB)]NAND[BNAND(ANANDB)]} MUX[edit] AmultiplexeroraMUXgateisathree-inputgatethatusesoneoftheinputs,calledtheselectorbit,toselectoneoftheothertwoinputs,calleddatabits,andoutputsonlytheselecteddatabit.[1] DesiredMUXGate NANDConstruction Q=[AANDNOT(S)]OR(BANDS) =[ANAND(SNANDS)]NAND(BNANDS) TruthTable InputA InputB Select OutputQ 0 0 0 0 0 1 0 0 1 0 0 1 1 1 0 1 0 0 1 0 0 1 1 1 1 0 1 0 1 1 1 1 DEMUX[edit] Ademultiplexerperformstheoppositefunctionofamultiplexer:Ittakesasingleinputandchannelsittooneoftwopossibleoutputsaccordingtoaselectorbitthatspecifieswhichoutputtochoose.[1] DesiredDEMUXGate NANDConstruction TruthTable Input Select OutputA OutputB 0 0 0 0 1 0 1 0 0 1 0 0 1 1 0 1 Seealso[edit] CMOStransistorstructuresandchipdepositiongeometriesthatproduceNANDlogicelements Shefferstroke–othername NORlogic.LikeNANDgates,NORgatesarealsouniversalgates. Functionalcompleteness Externallinks[edit] TTLNANDandANDgates-AllAboutCircuits StepstoDeriveXORfromNANDgate. NandGame-agameaboutbuildingacomputerusingonlyNANDgates References[edit] ^abNisan,N.&Schocken,S.,2005.In:FromNANDtoTetris:BuildingaModernComputerfromFirstPrinciples.s.l.:TheMITPress,p.20.Availableat:http://www.nand2tetris.org/chapters/chapter%2001.pdfArchived2017-01-10attheWaybackMachine Lancaster,Don(1974).TTLCookbook(1st ed.).Indianapolis,IN:HowardWSams.pp. 126–135.ISBN 0-672-21035-5. Sheffer,H.M.(1913),"AsetoffiveindependentpostulatesforBooleanalgebras,withapplicationtologicalconstants",TransactionsoftheAmericanMathematicalSociety,14:481–488,doi:10.2307/1988701,JSTOR 1988701 Retrievedfrom"https://en.wikipedia.org/w/index.php?title=NAND_logic&oldid=1089767213" Categories:LogicgatesHiddencategories:WebarchivetemplatewaybacklinksArticlesneedingadditionalreferencesfromJuly2012Allarticlesneedingadditionalreferences Navigationmenu Personaltools NotloggedinTalkContributionsCreateaccountLogin Namespaces ArticleTalk English Views ReadEditViewhistory More Search Navigation MainpageContentsCurrenteventsRandomarticleAboutWikipediaContactusDonate Contribute HelpLearntoeditCommunityportalRecentchangesUploadfile Tools WhatlinkshereRelatedchangesUploadfileSpecialpagesPermanentlinkPageinformationCitethispageWikidataitem Print/export DownloadasPDFPrintableversion Languages العربيةCatalàEspañolفارسیNorskbokmålRomână Editlinks