MINIMIZATION OF BOOLEAN FUNCTIONS 1423 



The first step in the revised method for determining prime implicants 

 is to list in a column, such as that shown in Table 11(a), the binary 

 equivalents of the decimal numbers which specify the function. It is 

 expedient to order these binary numbers so that any numbers which 

 contain no I's come first, followed by any numbers containing a single 

 1, etc. Lines should be drawn to divide the column into groups of binary 

 numbers which contain a given number of I's. The theorem stated above 

 is applied to these binary numbers by comparing each number with all 

 the numbers of the next lower group. Other pairs of numbers need not 

 be considered since any two numbers which are not from adjacent groups 

 must differ in more than one binary digit. For each number w^hich has 

 I's wherever the number (from the next upper group) with which it is 

 being compared has I's, a new character is formed according to the 

 theorem. A check mark is placed next to each number which is used in 

 forming a new character. The new characters are placed in a separate 

 column, such as Table 11(b), which is again divided into groups of char- 

 acters which have the same number of I's. The characters in this new 

 column will each contain one dash. 



After each number in the first column has been considered, a similar 

 process is carried out for the characters of column two. Two characters 

 from adjacent groups can be combined if they both have their dashes 

 ill the same position and if the character from the lower group has I's 

 wherever the upper character has I's. If any combinations are possible 

 the resulting characters are placed in a third column such as Table 11(c), 

 and the Column II characters from which the new characters are formed 

 are checked. All the characters in this third column will have two dashes. 

 This procedure is repeated and new columns are formed, Table 11(d), 

 until no further combinations are possible. The unchecked characters, 

 which have not entered into any combinations, represent the prime 

 implicants. 



Each binary character is labeled with the decimal equivalents of the 

 binary numbers which it represents (see note in Example 3.1). These 

 decimal numbers are arranged in increasing arithmetic order. For a 

 character having one dash this corresponds to the order of its formation : 

 When two binary numbers combine, the second number always contains 

 all the i's of the first number and one additional 1 so that the second 

 number is always greater than the first. Characters having two dashes 

 can be formed in two ways. For example, the character (0, 2, 4, 6) can 

 be formed either by combining (0, 2) and (4, 6) or by combining (0, 4) 

 and (2, 6) as given in Table III. Similarly, there are three ways in which 

 a character having three dashes can be formed (in Table II the 0, 2, 4, 



