1420 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 



nations, the resulting term will equal "one" when the variables have 

 values corresponding to either row i or row j of the table. Similarly, 

 when this theorem is used to replace, by a single term, a term which 

 equals "one" for rows i and j and a term which equals "one" for rows 

 k and m, the resulting term will equal "one" for rows i, j, k and m of the 

 table of combinations. A method for obtaining a minimum sum by re- 

 peated application of this theorem (.Ti.r2' + .Ti.'C2 = Xi) was first pre- 

 sented by Quine.^ In this method, the theorem is applied to all possible 

 pairs of p-terms, then to all possible pairs of the terms obtained from 

 the p-terms, and so on, until no further applications of the theorem are 

 possible. It may be necessary to pair one term with several other terms 

 in applying this theorem. In Example 3.2 the theorem is applied to the 

 terms labeled 5 and 7 and also to the terms labeled 5 and 13. All terms 

 paired with other terms in applying the theorem are then discarded. The 

 remaining terms are called prime implicants. Finally a minimum sum is 

 formed as the sum of the fewest prime implicants which when taken to- 

 gether will equal "one" for all required rows of the table of combinations. 

 The terms in the minimum sum will be called minimum sum terms or 

 ms-terms. 



Example 3.1 



T = Z(3, 7, 8, 9, 12, 13) 

 Canonical Expansion: 

 T = x/xi'xsXi -{- Xi'xiXsXi + a:ia;2'a^3 Xt + XyXz Xs Xi 



11 

 3 



111 



7 



10 



8 



10 1 

 9 



-f a:ia;2a;3'a;4' + XiX^Xs'xi 



110 

 12 



110 1 

 13 



The bracketed binary and decimal numbers below the sum terms indi- 

 cate the rows of the table of combinations for which the corresponding 

 term will equal "one." A binary character in Avhich a dash appears 

 represents the two binary numbers which are formed by replacing the 

 dash by a "0" and then by a "1." Similarly a binary character in which 

 two dashes appear represents the four binary numbers formed by re- 

 placing the dashes by "0" and "1" entries, etc. 



a;i'a;2 x^Xi + xi x^x^xt = xi x^xt 



11 

 3 







1 1 1 



7 



"O-l l1 

 _ 3,7 J 



