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DigitalFundamentalsEx1:4-1BooleanOperationsandExpressionsBooleanOperationsandExpressions4-2LawsandRulesofBooleanAlgebraLawsBooleanAlgebraLawsofBooleanAlgebraLawsofBooleanAlgebra求证:A+BC=(A+B)(A+C)RulesofBooleanAlgebraRulesofBooleanAlgebraRulesofBooleanAlgebra证明:混合变量的被吸收:RulesofBooleanAlgebra4-3DeMorgan’sTheorem(德·摩根定理)DeMorgan’sTheorems4-5SimplificationUsingBooleanAlgebra(利用布尔代数进行化简)Ex1:Ex2:conclusion:XORgatecanbeimplementedbyusingfourNANDgates.异或门可以用4个与非门实现:Ex4:simplifytheexpressionY。4-8TheKarnaughMap(卡诺图)SomeDefinitions(2)Maxterm最大项Minterm:Mintermcannotbebrokenupbecauseitcontainsallofthevariables.Wecanobtaintheexpressionfromthetruthtableaccordingtotheminterms.Celladjacency(逻辑相邻):thereisonlyasingle-variablechangebetweenadjacencycells.Adjacencycells将n个输入变量的全部最小项与输出量之间的关系用小方块阵列图表示,并且将逻辑相邻的最小项放在相邻的几何位置上,所得到的阵列图就是n变量的卡诺图。ABY 001 011 101 110Adjacencycell:thereisonlyasingle-variablechange。00F(A,B,C)=(m1,m2,m4,m7)Numberof4-variableKarnaughmap:SimplificationUsingKarnaughMapsAAWeshouldfollowthesesteps. Step1.寻找可合并的单元格(aslargeaspossible),将其圈入一个卡诺圈,圈得一个乘积项。相邻单元的个数是2n个,并组成矩形时,可以合并。c.将所有卡诺圈表示的乘积项逐个加起来,即得该函数的最简“与或”表示式。Example1:Example2:AB101状态未给出,即是无关项。化简时可以将无关项当作1或0,目的是得到最简结果。Tip3:圈完全部的卡诺圈后要检查一下有没有冗余。ASimplythefollowingKarnaughmaps:Howmanymeanscouldbeusedtorepresentthelogicfunction? Truthtable Logicexpression(Booleanexpression) Logiccircuit(symbol) Karnaughmap PulsedwaveformThischapteryoushouldmaster: