684 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1951 



contacts. To show that this number cannot be improved very much we will 

 now show that almost all switching functions require a number of contacts 

 of this order of magnitude. 



Theorem VIII. Let any €> be given. The fraction of switching functions 

 which can he synthesized using less than 



(13) (1-e) 2^^^^^^^) 



H+\og\og4>{N) 



contacts approaches zero uniformly as the number M of switches becomes large. ^ 

 Proof. The number of switching functions which can be constructed with 

 K contacts or less is certainly smaller than the number of ways the K 

 branches of the G{N, K) graphs can be replaced by contacts er{x^) or open 

 circuits; i.e. smaller than 



G{N, K) 



/ M \1 



By theorem VI the fraction F{K) of the {(l){N)Y switching functions which 

 can be built using K contacts or less satisfies 



(M \K 



E A + ij i't'Wr"' 



2N+k( logi5:+2+log 2 Pi+^ )-2^ log « 



where we have used • 



log2 (N + 2K) < loga 2K + ^ log, e 



<hS,2K+~. 



When K is the expression (13), one finds 



F{K) < 2''^(0(iV))' (-■ ^mo«-i^>--). 



Since --^_—^ — — °— -_^* approaches zero as the number of switches M gets 

 large, it follows that for sufficiently large M and any N 



• The word uniformly is used to indicate that the fraction in question can be made 

 smaller than any given number 6 > by making M larger than a certain number M{S) 

 which depends on S but not on N. 



