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5.3

SimplificationUsingKarnaughMapCIVILAVIATIONFLIGHTUNIVERSITYOFCHINA

TheKarnaughmap(K-map)techniquecanbeusedtosimplifytheBooleanfunctions.5.3.1KarnaughMapThe2-VariableK-Map

The3-VariableK-Map

CIVILAVIATIONFLIGHTUNIVERSITYOFCHINAAB0101ABc0100011110VariablesCellsBinarynumbersK-MapNotmorethan6variables.Someinformationrepresentedinthecells.Representationoftheinputvariablevalues.5.3.1KarnaughMapThe2-VariableK-Map

The3-VariableK-Map

CIVILAVIATIONFLIGHTUNIVERSITYOFCHINAAB0101ABc0100011110Cells:24cells38cellsn2ncells5.3.1KarnaughMapThe2-VariableK-Map

The3-VariableK-Map

CIVILAVIATIONFLIGHTUNIVERSITYOFCHINAAB0101ABc0100011110Binarynumbers:0100011110011100011011Thereisonlyasingle-variablevaluechangedbetweenadjacentcells.?5.3.1KarnaughMapThe2-VariableK-Map

The3-VariableK-Map

CIVILAVIATIONFLIGHTUNIVERSITYOFCHINAAB0101ABc0100011110865743214321Cell1’sadjacentcellsarecell2and3.

Whatarethecell4’sadjacentcells?

Cell3’sadjacentcellsarecell1,4and7.

Pleasegivecell1’sadjacentcells!

23174Cell2and3Cell2and3and523523ABc0100011110865743215.3.1KarnaughMapThe3-VariableK-Map

CIVILAVIATIONFLIGHTUNIVERSITYOFCHINAABc010001111086574321Theadjacentcellsincludethecellslocatedinsymmetricplace.

235145.3.1KarnaughMapThe4-VariableK-MapCIVILAVIATIONFLIGHTUNIVERSITYOFCHINAABCD0001110001111010124356789101211131615Pleasegivecell11’sadjacentcells!

Cell15,12Cell15,12,3and91512395.3.1KarnaughMapCIVILAVIATIONFLIGHTUNIVERSITYOFCHINA1.nvariables2n

Cells.2.Thereisonlyasingle-variablevaluechangedbetweenadjacentcells.3.Theadjacentcellsincludethecellslocatedinsymmetricplace.

14The4-VariableK-MapABCD00011100011110101243567891012111316155.3.2RepresentationofTruthTableonK-MapExample1:Accordingtothetruthtable,designaK-mapandrepresentthetruthtableonK-map.Solution:CIVILAVIATIONFLIGHTUNIVERSITYOFCHINATable5.3Truthtableofa3-variablefunctionInputOutputABCY00001111001100110101010101101001RowNo.123456780115.3.2RepresentationofTruthTableonK-MapExample1:CIVILAVIATIONFLIGHTUNIVERSITYOFCHINATable5.3Truthtableofa3-variablefunctionFig.5.3K-mapforTable5.3InputOutputABCY00001111001100110101010101101001RowNo.12345678ABc01000111100101023=8cells011Conclusion:allthevaluesofvariablesonK-maparethesameasthoseintruthtable,enteroutputvalueoftruthtableincorrespondingcell.5.3.3RepresentationofStandardSOPFormonK-MapExample2:RepresentthefollowinglogicalfunctiononK-Map.CIVILAVIATIONFLIGHTUNIVERSITYOFCHINANonstandardSOP(SumofProduct):Solution:ConvertthefunctionintostandardSOPform.Review:Minterm(StandardProductForm):Forexample:3variablesA,B,C.thereare23=8minterms:nvariables2nminterms5.3.3RepresentationofStandardSOPFormonK-MapExample2:RepresentthefollowinglogicalfunctiononK-Map.CIVILAVIATIONFLIGHTUNIVERSITYOFCHINANonstandardSOP(SumofProduct):Solution:ConvertthefunctionintostandardSOPform.Review:Minterm(StandardProductForm):ProductFormMintermnvariables2nminterms5.3.3RepresentationofStandardSOPFormonK-MapExample2:RepresentthefollowinglogicalfunctiononK-Map.CIVILAVIATIONFLIGHTUNIVERSITYOFCHINANonstandardSOP(SumofProduct):Solution:ConvertthefunctionintostandardSOPform.Review:StandardSOPForm:Forexample:?(instandardSOPForm)(notinstandardSOPForm)5.3.3RepresentationofStandardSOPFormonK-MapExample2:RepresentthefollowinglogicalfunctiononK-Map.CIVILAVIATIONFLIGHTUNIVERSITYOFCHINANonstandardSOP(SumofProduct):Solution:ConvertthefunctionintostandardSOPform.Thesameway:5.3.3RepresentationofStandardSOPFormonK-MapExample2:RepresentthefollowinglogicalfunctiononK-Map.CIVILAVIATIONFLIGHTUNIVERSITYOFCHINANonstandardSOP(SumofProduct):Solution:ConvertthefunctionintostandardSOPform.RepresenttheSOPformonK-Map.5.3.3RepresentationofStandardSOPFormonK-MapExample2:RepresentthefollowinglogicalfunctiononK-Map.CIVILAVIATIONFLIGHTUNIVERSITYOFCHINAFeatureofminterm:Solution:Fig.5.4K-mapABc0100011110A=0,B=0,C=0A=0,B=0,C=1A=0,B=1,C=05.3.3RepresentationofStandardSOPFormonK-MapExample2:RepresentthefollowinglogicalfunctiononK-Map.CIVILAVIATIONFLIGHTUNIVERSITYOFCHINASolution:Fig.5.4K-mapABc0100011110111115.3.3RepresentationofStandardSOPFormonK-MapExample2:RepresentthefollowinglogicalfunctiononK-Map.CIVILAVIATIONFLIGHTUNIVERSITYOFCHINASolution:Fig.5.4K-mapABc0100011110111115.3.4SimplificationofLogicalFunctionusingK-MapSimplificationtechniqueExample3:

simplifythelogicalfunctionusingK-map.CIVILAVIATIONFLIGHTUNIVERSITYOFCHINAStep1Markthosecellswitha1thatcorrespondtothetermsintheexpression.Fig.5.5K-mapABc010001111011111123758465.3.4SimplificationofLogicalFunctionusingK-MapSimplificationtechniqueCIVILAVIATIONFLIGHTUNIVERSITYOFCHINAStep2Formtheadjacent1’sintothelargestvalidgroupwhichmustbearectangleandmustcontain2i(i=0,1,...,n)cells.Fig.5.5K-mapABc010001111011111Rectangle;20=1cells;Butnotthelargest??Rectangle;21=2cells;Butnotthelargest123758465.3.4SimplificationofLogicalFunctionusingK-MapSimplificationtechniqueCIVILAVIATIONFLIGHTUNIVERSITYOFCHINAStep2Formtheadjacent1’sintothelargestvalidgroupwhichmustbearectangleandmustcontain2i(i=0,1,...,n)cells.Fig.5.5K-mapRectangle;22=4cells;ThelargestABc01000111101111112375846ABc010001111011111123758465.3.4SimplificationofLogicalFunctionusingK-MapSimplificationtechniqueCIVILAVIATIONFLIGHTUNIVERSITYOFCHINAStep2Formtheadjacent1’sintothelargestvalidgroupwhichmustbearectangleandmustcontain2i(i=0,1,...,n)cells.Fig.5.5K-mapNotaRectangle;5cells;ABc01000111101111112375846ABc01000111101111112375846?5.3.4SimplificationofLogicalFunctionusingK-MapSimplificationtechniqueCIVILAVIATIONFLIGHTUNIVERSITYOFCHINAStep2Formtheadjacent1’sintothelargestvalidgroupwhichmustbearectangleandmustcontain2i(i=0,1,...,n)cells.Fig.5.5K-mapRectangle;3cells;ABc01000111101111112375846ABc01000111101111112375846?5.3.4SimplificationofLogicalFunctionusingK-MapSimplificationtechniqueCIVILAVIATIONFLIGHTUNIVERSITYOFCHINAStep2Formtheadjacent1’sintothelargestvalidgroupwhichmustbearectangleandmustcontain2i(i=0,1,...,n)cells.Fig.5.5K-mapRectangle;21=2cells;ABc01000111101111112375846ABc010001111011111123758465.3.4SimplificationofLogicalFunctionusingK-MapSimplificationtechniqueCIVILAVIATIONFLIGHTUNIVERSITYOFCHINAStep2Formtheadjacent1’sintothelargestvalidgroupwhichmustbearectangleandmustcontain2i(i=0,1,...,n)cells.Fig.5.5K-mapABc01000111101111112375846ABc010001111011111123758465.3.4SimplificationofLogicalFunctionusingK-MapSimplificationtechniqueCIVILAVIATIONFLIGHTUNIVERSITYOFCHINAStep2Formtheadjacent1’sintothelargestvalidgroupwhichmustbearectangleandmustcontain2i(i=0,1,...,n)cells.Fig.5.5K-mapABc0100011110111115.3.4SimplificationofLogicalFunctionusingK-MapSimplificationtechniqueCIVILAVIATIONFLIGHTUNIVERSITYOFCHINAStep3

Each1onthemapmustbeincludedinatleastonegroup.The1salreadyinagroupcanbeincludedinanothergroupaslongastheoverlappinggroupsincludenoncommon1s.Fig.5.5K-mapABc01000111101111115.3.4SimplificationofLogicalFunctionusingK-MapSimplificationtechniqueCIVILAVIATIONFLIGHTUNIVERSITYOFCHINAStep4Identifyadjacentones,thenseethevaluesofthevariableassociatedwiththesecells.Ifthevariableswill

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