SYNTHESIS OF SWITCH I XG CIRCUITS 67 



described as follows: The function to be realized is written in the form of a 

 product of the type (1) in such a way that the functions Uk are the same for a 

 large class of functions, the Vk determining the particular one under consider- 

 ation. A basic disjunctive network M is constructed having the functions 

 Uk between terminals a and k. A network N for obtaining the functions 

 Vk is then found by inspection or according to certain general rules. We 

 will now consider just how this can be done in various cases. 



3. Design of Networks for General Functions — Beh.a.vior of X(w)- 

 a. Functions of One, Two and Three Variables: 



Functions of one or two variables may be dismissed easily since the 

 number of such functions is so small. Thus, with one variable X, the 

 possible functions are only: 



0, 1, X, X' 



and obviously X(l) = 1, m(1) = 1- 



With two variables X and F there are 16 possible functions: 



X F XY XV X'Y X'Y' XY' + X'Y 



\X' Y' X + F X+Y' X' +Y X' + F' XF + X'Y' 



so that X(2) = 4, m(2) = 2. 



We will next show that any function of three variables /(X, F, Z) can be 

 realized with not more than eight elements and with not more than four 

 from any one relay. Any function of three variables can be expanded in a 

 product as follows: 



/(X, F, Z) = [X + F + /(O, 0, Z)][X + F' + /(O, 1, Z)\ 



[X' + F + /(I, 0, Z)] [X' + F' + /(I, 1, Z)]. 



In the terminology of Theorem 1 we let 



f/i = X + F Ki = /(O, 0, Z) 



U,= X+Y' V,^ /(O, 1, Z) 



f/3 = X' + F F3 = /(I, 0, Z) 



f/4 = X' + Y' V, = /(I, 1, Z) 



so that 



4 



f'a6 =/(X, F,Z) = ]l{Vk+ Vk) 

 k-X 



The above Uk functions are realized with the network M of Fig. 7 and it is 



