boolean algebra and logic gates chapter 2 topicscourses.ece.ubc.ca/256/lectures/2010/digital logic...

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BooleanAlgebraandLogicGatesChapter2

EECE256Dr.SidneyFels

StevenOldridge

Topics

•  DefiniGonsofBooleanAlgebra•  AxiomsandTheoremsofBooleanAlgebra–  twovaluedBooleanAlgebra

•  BooleanFuncGons– simplificaGon

•  Canonicalforms– mintermandmaxterms

•  Otherlogicgates9/18/10 2(c)S.Fels,since2010

BooleanAlgebra

•  AllowsustodefineandsimplifyfuncGonsofbinaryvariables

•  Importantfordesignerstocreatecomplexcircuits–  funcGonsofcomputer

– ASICdevices– programmablelogic– determinemachinestatetransiGons

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BooleanAlgebra

•  Adherestothelawsofanalgebra– closure– associaGve– commutaGve–  idenGty–  inverse– distribuGve+foraddiGon(0isidenGty)formulGplicaGon(1isidenGty)

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AxiomsofBooleanAlgebra

•  closurefor+and•  IdenGty:

x+0=x1=

•  commutaGvex+y=xy=yx

•  distribuGvex(y+z)=x+(yz)=

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AxiomsofBooleanAlgebra

•  Complement–  x+x’=1xx’=0

•  twoelementsforTwo‐ValuedBooleanAlgebra0and1;0!=1AND=,OR=+,NOT=inverse‐  checkwithTruthtablesandyou’llseeitmeetsalltheaxioms

•  switchingalgebra(Shannon,1928)–  basisofalldigitalcomputers

•  Precedence:–  parentheses,NOT,AND,OR

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TheoremsandProperGesofBooleanAlgebra

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idempotent0and1ops

complementidenGty

TheoremsandProperGesofBooleanAlgebra

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Duality:interchange0for1andANDandOR

idempotent0and1ops

complementidenGty

TheoremsandProperGesofBooleanAlgebra

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Duality:interchange0for1andANDandOR

idempotent0and1ops

complementidenGty

TheoremsusedtosimplifycomplexfuncGonsofbinaryvariables

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UsefulTheorems

•  SimplificaGonTheorems:–  XY+XY’=–  X+XY=–  (X+Y’)Y=

•  DeMorgan’sLaw:–  (X+Y)’=

•  TheoremsforMulGplyingandFactoring:–  (X+Y)(X’+Z)=XZ+X’Y

•  Proofsbyalgebracomplicated–  usetruthtablesinstead

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Somealgebraicproofs

ProvingTheoremsviaaxiomsofBooleanAlgebra:

e.g.,Prove:XY+XY’=X

e.g.,Prove:X+XY=X

e.g.,Prove:(X+Y)(X’+Z)=XZ+X’Y

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SomealgebraicproofsProvingTheoremsviaaxiomsofBooleanAlgebra:

e.g.,Prove:XY+XY’=XLHS=X(Y+Y’)distribuGve =X(1)complement =X=RHSidenGty

e.g.,Prove:X+XY=X LHS=X(1+Y)distribuGve =X(1) idenGty =X=RHS

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Somealgebraicproofs

e.g.,Prove:(X+Y)(X’+Z)=XZ+X’Y LHS=(X+Y)X’+(X+Y)ZdistribuGve

=XX’+YX’+XZ+YZdistribuGve

=0+X’Y+XZ+YZcomplement,associaGve,distribuGve

=X’Y(Z+Z’)+XZ(Y+Y’)+YZ(X+X’)idenGty/complement

=X’YZ+X’YZ’+XYZ+XY’Z+XYZ+X’YZdistribuGve,associaGve

=XZ(Y+Y’)+X’Y(Z+Z’)idempotent,associaGve,distribuGve

=XZ+X’Y=RHScomplement

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Someproofsusingtruthtables

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DeMorgan’sLaw

(X+Y)’=X’Y’

(XY)’=X’+Y’

X Y X’ Y’ (X+Y) (X+Y)’ X’*Y’

0 0 1 1

0 1 1 0

1 0 0 1

1 1 0 0

X Y X’ Y’ (X*Y) (X*Y)’ X’+Y’

0 0 1 1

0 1 1 0

1 0 0 1

1 1 0 0

Someproofsusingtruthtables

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DeMorgan’sLaw

(X+Y)’=X’Y’

(XY)’=X’+Y’

X Y X’ Y’ (X+Y) (X+Y)’ X’*Y’

0 0 1 1 0 1 1

0 1 1 0 1 0 0

1 0 0 1 1 0 0

1 1 0 0 1 0 0

X Y X’ Y’ (X*Y) (X*Y)’ X’+Y’

0 0 1 1 0 1 1

0 1 1 0 0 1 1

1 0 0 1 0 1 1

1 1 0 0 1 0 0

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DeMorgan’sThereom

Example:Z=A’B’C+A’BC+AB’C+ABC’

Z’=(A+B+C’)(A+B’+C’)(A’…..

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BooleanFuncGons

•  Now,wehaveeverythingtomakeBooleanFuncGons– F=f(x,y,z…)wherex,y,zetc.arebinaryvalues(0,1)withBooleanoperators

– circuitscanimplementthefuncGon– algebrausedtosimplifythefuncGontomakeiteasiertoimplement

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Example•  F1=x+y’z

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x y z y’z x+y’z

0 0 0

0 0 1

0 1 0

0 1 1

1 0 0

1 0 1

1 1 0

1 1 1

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Example•  F1=x+y’z

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x y z y’z x+y’z

0 0 0 0 0

0 0 1 1 1

0 1 0 0 0

0 1 1 0 0

1 0 0 0 1

1 0 1 1 1

1 1 0 0 1

1 1 1 0 1

SimplificaGonallowsfordifferentimplementaGons

•  F=AB+C(D+E)=requires3levelsofgates

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2‐levelimplementaGon

•  F=AB+C(D+E)=AB+CD+CE

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CanonicalForms

•  ExpressallBooleanfuncGonsasoneoftwocanonicalforms– enumeratesallcombinaGonsofvariablesaseither•  SumofProducts,i.e.,m1+m2+m3…etc•  ProductofSums,i.e.,M1M2M3…etc

– eachvariableappearsinnormalform(x)oritscomplement(x’)

–  ifitisaproductitiscalledaMINTERM–  ifisisasumitiscalledaMAXTERM– nvariables‐>2nMINTERMSorMAXTERMS

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CanonicalForms

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CanonicalFormExample

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CanonicalFormExample

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m1

m4

m7

m3

m5

m6

m7

CanonicalFormExample:SumofProducts(Minterms)

•  SowecanreadoffofTTdirectly•  SumofproductsissumofMinterms

F1=m1+m4+m7 =x’y’z+xy’z’+xyz

F2 =m3+m5+m6+m7 =x’yz+xy’z+xyz’+xyz

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€

mi∑

CanonicalFormExample

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M0

M2

M5

M6

M3

M0

M1

M4

M2

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CanonicalFormExample:ProductofSums(Maxterms)

•  SowecanreadoffofTTdirectly•  ProductofsumsisproductofMaxterms

F1=M0M2M3M5M6=(x+y+z)(x+y’+z)(x+y’+z’)(x’+y+z’)(x’+y’+z)

F2=M0M1M2M4=(x+y+z)(x+y+z’)(x+y’+z)(x’+y+z)

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€

Mi∏

ConverGngbetweenthem

•  YoucanusecomplementanddeMorgan’stheorem–  ifF=m1+m3+m5i.e.Σ(1,3,5)then– F’=m0+m2+m4+m6+m7•  F=(m0+m2+m4+m6+m7)’

– useDeMorgan’snowtogetProductofSum•  F=Π(0,2,4,6,7)

•  RemembertoincludeallMinterms/Maxterms– nvariables,2nterms

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Standardform•  SumofProductswithone,two,threeormorevariablesinproduct

form–  F1=y’+xy+x’yz’

•  ProductofSumwithone,two,threeormorevariablesinsumform–  F2=x(y’+z)(x’+y+z’)

•  NoGce:canonicalandstandardformare2‐levelimplementaGons–  butmayhavemanyinputsforgate

•  calledfan‐in;limitedbypinsonICandmanufacturing

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OtherlogicaloperaGons

•  for2inputgates,youcanhave16differentlogicoperaGons22

nwheren=2

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OtherlogicaloperaGons

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DigitalLogicGates

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ExtendingtomulGpleinputs

•  worksfinefor– AND,OR;noproblem–commuteandassociate– NAND,NOR–commutebutdon’tassociate•  so,becarefulwhenusingthemcascaded•  instead:

–  definemulG‐inputNANDasmulG‐inputANDthatisinvertedattheend»  xNANDyNANDz=(xyz)’

–  definemulG‐inputNORasmulG‐inputORthatisinvertedattheend»  xNORyNORZ=(x+y+z)’

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ExtendingtomulGpleinputs

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ExtendingtomulGpleinputs

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Summary

•  two‐valuedBooleanalgebrasupports–  switchinglogic–  simplificaGonpostulatesandtheorems–  digitallogicgates

•  TruthtablescanbeusedtodefinefuncGon•  CanonicalandstandardformsmakeiteasytocreatefuncGonsthatcanbeimplemented

•  finitenumberof2inputgates–  easytoimplementlargercomplexfuncGons

•  2inputgatescanbeextendedtomulGpleinputs

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