62 BELL SYSTEM TECHNICAL JOURNAL 



called a. Junction of these variables and written in ordinary function notation, 

 f{X, , X2 , • • • , Xn). Thus we might have /(X, F, Z) = X + I"Z + XZ'. 

 In Boolean Algebra there are a number of important general theorems which 

 hold for any function. It is possible to expand a function about one or more 

 of its arguments as follows: 



/(Xi , X2 , • • • , X„) = X:/(l, X2 , • • • , X„) + X7(0, X2 , • • • , Xn) 

 This is an expansion about Xi . The term /(I, X2 , • • • , X„) is the function 



-• = •- 



X + X + X 





(X + Y) (X + Z) 



-c;:> = 



XX' = 



Fig. 3 — Interpretation of some special Boolean identities. 



/(Xi , X2 , • ■ • , X„) with 1 substituted for A^, and for X', and conversely 

 for the term/(0, X2 , • • • , X„). An expansion about X'': and A'2 is: 



/(Xi,X2,--- ,X„) = X:X2/(l,l,X3,---,X„) + XiX2/(l,0,X3,---,X„) 



+ X(X2/(0, 1, X3 , • • • , X„) + X(X2/(1, 1, X3 , • • • , X„) 



This may be continued to give expansions about any number of variables. 

 When carried out for all n variables, / is written as a sum of 2" products 

 each with a coefficient which does not depend on any of the variables. 

 Each coefficient is therefore a constant, either or 1. 



There is a similar expansion whereby/ is expanded as a product: 



/(Xi,X2,---,X2) 



= [Xi + /(O, X2 , • • • , X„)] [X( + /(I, X2 , • • • , X„)] 



= [Xi+ X2+/(0,0, • • • ,X,0] [Xi + X2+/(0, 1, • • • ,X„)] 



[x; + X2 + /(1, 0, • • • , X,.)] [x; + Xo + /(1, 1, • • • . Xn)\ 



= etc. 



