TV-TERMINAL SWITCHING CIRCUITS 



671 



There is no simple identity which corresponds to the Boolean Algebra 

 expansion by sums. An identity analogous to the Boolean Algebra expansion 

 about xi by products is 



(1) 



»=1 



where the range of xi is from 1 to p. To prove (1) we need only observe that 

 in the product all the terms for which i 9^ Xi have the value 1. The remain- 

 ing term, for which i = Xi , has the value /(xi , • • • , Xm)- The switching 

 interpretation of (1) is illustrated in Fig. 2. By repeated use of (1) it follows 

 that any function f{xi , • • • , Xm) can be written as an expression involving 



Fig. 2 — Expansion oijixx , . . . , xm) about x\ . 



parentheses, addition signs, multipUcation signs, the ^ife), and nothing 

 else. Such expressions may be regarded as Boolean functions with the eiix^ 

 as variables; they may be rearranged and factored according to the usual rules 

 of Boolean Algebra. However, one should keep in mind that the ei{x^ are 

 subject to the constraints that a selector switch can be in only one position 

 at any given time. The effect of these constraints is to add a cancellation law 



6/.fe) + ^tfe) = 1 \i ]i7^ i. 

 The inverse ei{xj) of the Boolean variable ei{xj) is the Boolean function 



ei{xj) = 



1 when ei{Xj) = 

 when ei{xj) = 1 . 



