.,1,chapter4,CombinationalLogicDesignPrinciples,.,2,chapter4SwitchingAlgebraCombinationalCircuitAnalysisCombinationalCircuitSynthesisTimingHazards,.,3,BasicConcept(基本概念),Logiccircuitsareclassifiedintotwotypes（邏輯電路分為兩大類）combinationallogiccircuit（組合邏輯電路）Acombinationallogiccircuitisonewhoseoutputsdependonlyonitscurrentinputs.（任何時刻的輸出僅取決與當(dāng)時的輸入）characteristic：nofeedbackcircuitsequentiallogiccircuit（時序邏輯電路）Theoutputsofasequentiallogiccircuitdependnotonlyonthecurrentinputs,butalsoonthepastsequenceofinputs,possiblyarbitrarilyfarbackintime.（任一時刻的輸出不僅取決于當(dāng)時的輸入，還取決于過去的輸入順序）,.,4,4.1SwitchingAlgebra,4.1.1Axioms（公理）P185（A1）X=0ifX1,（A1）X=1ifX0（A2）IfX=0,thenX=1（A2）IfX=1,thenX=0（A3）00=0（A3）1+1=1（A4）11=1（A4）0+0=0（A5）01=10=0（A5）1+0=0+1=1,Westatedtheseaxiomsasapair,withtheonlydifferencebetweenA1andA1beingtheinterchangeofthesymbols0and1.Thisisacharacteristicofalltheaxiomsofswitchingalgebra.P（185）,邏輯乘logicalmultiplicationdot乘點(diǎn)multiplicationdot,.,5,4.1.2Single-VariableTheorems(單變量開關(guān)代數(shù)定理)P188,Identities(自等律)：(T1)X+0=X(T1)X1=XNullElements(0-1律)：(T2)X+1=1(T2)X0=0Idempotency(同一律):(T3)X+X=X(T3)XX=XInvolution(還原律)：(T4)(X)=XComplements(互補(bǔ)律):(T5)X+X=1(T5)XX=0,變量和常量的關(guān)系,變量和其自身的關(guān)系,.,6,4.1.3Two-andThree-VariableTheorems(1),Commutativity(交換律)(T6)X+Y=Y+X(T6)XY=YXAssociativity(結(jié)合律)(T7)X(YZ)=(XY)Z(T7)X+(Y+Z)=(X+Y)+ZDistributivity(分配律)(T8)X(Y+Z)=XY+XZ(T8)X+YZ=(X+Y)(X+Z),Eachofthesetheoremsiseasilyprovedbyperfectinduction.(可以利用完備歸納法證明公式和定理)P188SimilarRelationshipwithGeneralAlgebra(與普通代數(shù)相似的關(guān)系),.,7,4.1.3Two-andThree-VariableTheorems(2),Covering(吸收律)(T9)X+XY=X(T9)X(X+Y)=XCombining(合并律)(T10)XY+XY=X(T10)(X+Y)(X+Y)=XConsensus（添加律（一致性定理）(T11)XY+XZ+YZ=XY+XZ(T11)(X+Y)(X+Z)(Y+Z)=(X+Y)(X+Z),.,8,Notes,nopowerofnumber(沒有變量的乘方)AAAA3commonfactor(允許提取公因子)AB+AC=A(B+C)nodivision(沒有定義除法)ifAB=BCA=C?,Nosubtracting(沒有定義減法)ifA+B=A+CB=C?,A=1,B=0,C=0AB=BC=0,AC,A=1,B=0,C=1,錯！,錯！,.,9,4.1.4n-VariableTheorems(n變量定理),Generalizedidempotency(廣義同一律)(T12)X+X+X=X(T12)XXX=XDeMorgansTheorems(德.摩根定理)(T13)(X1X2Xn)=X1+X2+Xn(T13)(X1+X2+Xn)=X1X2XnGeneralizedDeMorgansTheorems(廣義德.摩根定理)(T14)F(X1,X2,Xn,+,)=F(X1,X2,Xn,+),Mostofthesetheoremscanbeprovedusingatwo-stepmethodcalledfiniteinductionfirstprovingthatthetheoremistrueforn=2(thebasisstep)andthenprovingthatifthetheoremistrueforn=i,thenitisalsotrueforn=i+1(theinductionstep).P(190),.,10,finiteinduction(P190),X+X+X+X=X+(X+X+X)(i+1Xsoneitherside)=X+(X)(ifT12istrueforn=i)=X(accordingtoT3),.,11,DemorgansTheorems(摩根定理)(P191),.,12,反演規(guī)則(ComplementRules)：swapping+and.andcomplementingallvariables.+，01，變量取反遵循原來的運(yùn)算優(yōu)先（Priority）次序不屬于單個變量上的反號應(yīng)保留不變,complementofalogicexpression(F)(反演定理)(P192),.,13,例1：寫出下面函數(shù)的反函數(shù)（Complementfunction)F1=A(B+C)+CDF2=(AB)+CDE,例2：證明(AB+AC)=AB+AC,合理地運(yùn)用反演定理能夠?qū)⒁恍﹩栴}簡化,.,14,合理地運(yùn)用反演定理能夠?qū)⒁恍﹩栴}簡化,(AB+AC),AB+AC+BC=AB+AC,(A+B)(A+C),AA+AC+AB+BC,AC+AB,AC+AB+BC,Example2：prove(AB+AC)=AB+AC,.,15,4.1.5duality(對偶定理)(P193),FD(X1,X2,Xn,+,)=F(X1,X2,Xn,+,),PrincipleofDualityAnytheoremoridentityinswitchingalgebraremainstrueif0and1areswappedand.and+areswappedthroughout.,thedualofalogicexpression,F(X1,X2,Xn)=FD(X1,X2,Xn),.,16,4.1.5duality(對偶定理)(P193),對偶規(guī)則+；01變換時不能破壞原來的運(yùn)算順序（優(yōu)先級）對偶原理(PrincipleofDuality)若兩邏輯式相等，則它們的對偶式也相等,例：寫出下面函數(shù)的對偶函數(shù)F1=A+B(C+D)F2=(A(B+C)+(C+D),X+XY=XXX+Y=X(錯),X(X+Y)=X,FD(X1,X2,Xn,+,)=F(X1,X2,Xn,+,),.,17,對偶定理（DualityTheorems）,證明公式：A+BC=(A+B)(A+C),.,18,DualityandComplement(對偶和反演)(P194.P195),對偶(Duality)：FD(X1,X2,Xn,+,)=F(X1,X2,Xn,+,),反演(Complement)：F(X1,X2,Xn,+,)=F(X1,X2,Xn,+),F(X1,X2,Xn)=FD(X1,X2,Xn),正邏輯約定和負(fù)邏輯約定互為對偶關(guān)系,.,19,ElectricalFunctionTable(電氣功能表),Positive-LogicConvention,Negative-LogicConvention,Positive-Logic(正邏輯)：F=AB,Negative-Logic(負(fù)邏輯)：F=A+B,TherelationshipofPositive-LogicConventionandNegative-LogicConventionareDuality(正邏輯約定和負(fù)邏輯約定互為對偶關(guān)系),.,20,.,21,Shannonsexpansiontheorems(香農(nóng)展開定理),香農(nóng)展開定理主要用于證明等式或展開函數(shù),將函數(shù)展開一次可以使函數(shù)內(nèi)部的變量數(shù)從n個減少到n-1個.,.,22,Shannonsexpansiontheorems,F=X.Y+Y.Z=X(1.Y+Y.Z)+X(0.Y+Y.Z)+Y(1.X+1.Z)+Y(0.X+0.Z)+Z(X.Y+Y.1)+Z(X.Y+Y.0)=X.Y.Z+X.Y.Z+X.Y.Z,.,23,舉重裁判電路,Y=F(A,B,C)=A(B+C),開關(guān)ABC1表閉合指示燈1表亮,4.1.6StandardRepresentationsofLogicFunctions(邏輯函數(shù)的標(biāo)準(zhǔn)表示方法),00000111,LogicFunctions,.,24,波形圖(WaveForm),將輸出與輸入信號變化的時間關(guān)系用波形的形式描述，就得到了波形圖,.,25,Truthtable真值表productterm乘積項(xiàng)sumterm求和項(xiàng)Asum-of-productsexpression“積之和”表達(dá)式product-of-sumsexpression“和之積”表達(dá)式n-variablemintermn變量最小項(xiàng)n-variablemaxtermn變量最