74 



BF.LL SYSTF.M TECHNICAL JOURNAL 



where /a and/b involve only m variables. By connecting from g to the cor- 

 responding fa and fb terminals of the smaller network, as shown typically 

 for ^3 , we see from Theorem 1 that all the g functions can be obtained. 

 Among these will be the 2'"' f functions and these can be obtained simply 

 by connecting across to the / functions in question without any additional 

 elements. Thus the entire network uses less than 



(2^ 



- l-"")! + 2-t 



elements, since the N network by assumption uses less than 2 • 2" and the 

 first term in this expression is the number of added elements. 



The second statement of Theorem 7 can be proved as follows. Suppose 

 we have a network, Fig. 16, with the required property. The terminals 

 can be divided into three classes, those that have one or less elements di- 



►fa 



.f,2"l 



Fig. 16 — NetWork giving all functions of m variables. 



rectly conriected, those with two, and those with three or more, The first 

 set consists of the functions and 1 and functions of the type 



(X+/) = X+/x=o 



where X is some variable or primed variable. The number of such functions 

 is not greater than Im-l^"" for there are 2m ways of selecting an "X" 

 and then 2"'" different functions /x=o of the remaining w — 1 variables. 

 Hence the terminals in this class as a fraction of the total -^ as w -^ <» . 

 Functions of the second class have the form 



g= (X+/0(F+/2) 



In case X 9^ V this may be written 



XY + XY'gx^i.y^o + X'Ygx=o.y=i + X'Y'g^^o.v^o 



and there are not more than (2m)(2m — 2)[2"'" f such functions, again a 

 vanishingly small fraction. In case X = Y' we have the situation shown 

 in Fig. 17 and the XX' connection can never carry ground to another 

 terminal since it is always open as a series combination. The inner ends 

 of these elements can therefore be removed and connected to terminals 



