68 BELL SYSTEM TECHNICAL JOURNAL 



easily seen that U jk — \ {j, k = 1,2, 3, 4). The problem now is to construct 

 a second network iV having the Vh functions Vi , V2 , V3 , Vi . Each of 

 these is a function of the one variable Z and must, therefore, be one of the 

 four possible functions of one variable : 



0, 1, Z, Z'. 



Consider the network N of Fig. 8. If any of the F's are equal to 0, connect 

 the corresponding terminals of M to the terminal of N marked 0; if any are 

 equal to Z, connect these terminals of M to the terminal of N marked Z, 

 etc. Those which are 1 are, of course, not connected to anything. It is 

 clear from Theorem 1 that the network thus obtained will realize the function 

 /"(X, F, Z). In many cases some of the elements will be superfluous, e.g., 

 if one of the F,- is equal to 1, the element of M connected to terminal i can 



Fig. 7 — Disjunctive tree with two bays. 



be eliminated. At worst M contains six elements and N contains two. 

 The variable X appears twice, F four times and Z twice. Of course, it is 

 completely arbitrary which variables we call X, F, and Z. We have thus 

 proved somewhat more than we stated above, namely. 



Theorem 2: Any function of three variables may be realized using not more 

 than 2, 2, and 4 elements from the three variables in any desired order. Thus 

 X(3) < 8, m(3) < 4. Further, since make and break elements appear in 

 adjacent pairs we can obtain the distribution 1, 1, 2, in terms of transfer ele- 

 ments. 



The theorem gives only upper limits for X(3) and n{i). The question 

 immediately arises as to whether by some other design method these limits 

 could be lowered, i.e., can the < signs be replaced by < signs. It can be 

 shown by a study of special cases that X(3) = 8, the function 



X @ Y ® Z^ X{YZ + Y'Z') + X' {YZ' + Y'Z) 



requiring eight elements in its most economical realization. m(3), however, 

 is actually 3. 



