SINTIIESIS OF SWITCHING CIRCUITS 



61 



« o"o- 



-o o- 

 X 



Y + X 



X (Y2) 



(XY) 2 



-0 o- 

 -o^o- 

 XY2 



hC 



^> 



X (Y+Z) = XY *• XZ 



Fig. 2— Interpretation of some algebraic identities. 



simplifications of expressions that are not possible in ordinary algebra. 

 The more important of these are: 



X = X+X = X^- X + X = etc. 



X= X-X = X-X-X = etc. 



X + 1 = 1 



X+YZ= {X+ Y){X+ Z) 



X + X' = 1 



X-X' = 



(X+ Y)' = X'Y' 



{XY)' = X' + r 



The circuit interpretation of some of these is shown in Fig. 3. These rules 

 make the manipulation of Boolean expressions considerably simpler than 

 ordinary algebra. There is no need, for example, for numerical coefficients 

 or for exponents, since nX = X" = X. 



By means of Boolean Algebra it is possible to find many circuits equivalent 

 in operating characteristics to a given circuit. The hindrance of the given 

 circuit is written down and manipulated according to the rules. Each 

 different resulting expression represents a new circuit equivalent to the given 

 one. In particular, expressions may be manipulated to eliminate elements 

 which are unnecessary, resulting in simple circuits. 



Any expression involving a number of variables Xi , X2 , • • • , X„ is 



