1440 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 



theorem (A + B)(A + C) = (A + BC) is useful. This example shows 

 that for a general table the expressions described in this Section and 

 the multipHcation process can become very lengthy. However, this pro- 

 cedure is entirely systematic and may be suitable for mechanization. 



Since the product of factors representation of a prime implicant 

 table is a Boolean expression, it can be interpreted as the transmission 

 of a contact network. Each consistent row set then corresponds to a 

 path through this equivalent network. By sketching the network directly 

 from the product of factors expression, it is possible to avoid the multi- 

 plication process. In particular the network should be sketched in the 

 form of a tree, as in Table XIII (c) and the Boolean Algebra theorems 

 used to simplify it as it is being drawn. For hand calculations, this 

 method is sometimes easier than direct multiplication. 



I 



10 d-TERMS 



In Section 1 the possibility of having rf-entries in a table of combina- 

 tions was mentioned. Whenever there are combinations of the relay 

 conditions for which the transmission is not specified, f/-entries are placed 

 in the T-column of the corresponding rows of the table of combinations. 



Table XIV — Determination of the Minimum 

 Sum for the Transmission 



T = X)(5, 6, 13) + f/(9, 14) Where 9 and 14 are the cI-Terms, 



(d) 



(a) Determination of Prime Implicants 



Xi Xs X2X1 



5 1 1 V 



6 1 1 V 

 9 1 1 V 



5 13 



6 14 

 9 13 



13 1 1 1 V 

 (d) 14 1 1 1 V 



(b) Prime Implicant Table 

 5 6 13 



X4X3 ^"2X1 



- 1 1 



- 1 1 

 1-01 



(c) 

 Basis rows: (5, 13), (6, 14) 



(d) 

 T = XaXi'xi + Tso-oa-i' 



