Digital Logical Circuit（俞鹤伟）chapter4-a.pptVIP

An n-input AND gate whose output is complemented is equivalent to an n-input OR gate whose inputs are complemented. DeMorgan’s theorems T13 T14 : Applies to an arbitrary logic expression F. The complement of a logic expression, denoted (F)′, is an logic expression whose independent variables are complement of the originals, AND become OR. Given any n-variable logic expression, its complement can be obtained by swapping + and · and complementing all variables. Generalized DeMorgan’s Law (14) DeMorgan’s Law (example) Express the complement f’(w,x,y,z) of the following expression in a simplified form. f(w,x,y,z) = wx(y’z + yz’) f’(w,x,y,z) = w’ + x’ + (y’z +yz’)’ = w’ + x’ + (y’z)’(yz’)’ = w’ + x’ + (y + z’)(y’ + z) = w’ + x’ + yy’ + yz + z’y’ + z’z = w’ + x’ + 0 + yz + z’y’ + 0 = w’ + x’ + yz + y’z’ * Practice about switching-algebra 1. Prove the following equations using switching-algebra : (a) A+A’B=A+B (b) ABC+AB’C+ABC’=AB+AC 2. Using switching-algebra theorems to simplify each of the following logic functions: (a) F= (AB’+ABC)’+A(B+AB’) (b) F= (A’+B)’+(A+B’)’+(A’B)’(AB’)’ Solution 1. (a) A+A’B=A+B solution：left side= A+A’B =A(B+B’)+A’B =AB+AB’+A’B =AB+AB’+AB+A’B =A+B =right side (b) ABC+AB’C+ABC’=AB+AC solution：left side= ABC+AB’C+ABC’ = ABC+AB’C+ABC’+ABC =AC(B+B’)+AB(C+C’) =AB+AC =right side Solution 2. (a) F= (AB’+ABC)’+A(B+AB’) solution：F=(AB’)’(ABC)’+A(B+AB’) =(A’+B)(A’+B’+C’)+AB+AB’ =A’+A’B’+A’C’+A’B+0+BC’+A =1 (b) F= (A’+B)’+(A+B’)’+(A’B)’(AB’)’ solution：F=AB’+A’B+(A+B’)(A’+B) =AB’+A’B+0+AB+A’B’+0 =A+A’ =1 4.1.5 Duality Any theore