
9.11 Minimal Expressions of Incomplete Boolean Functions

Incomplete Boolean expressions are those which evaluates to a don’t care condition for some input combinations. We wrote a in the truth table in section 1.8 for those conditions for which the function evaluates to ‘don’t care’. Karmargh maps of incomplete Boolean expressions would have cells with ‘’ entries besides cells in Is and Os. the cells with ‘’ are the don’t carecells just as the cells wi;h Is were called 1cells and those with Os were called 0cells. The prime implicants obtained by considering the don’t carecells as Icells are the prime implicants of an incomplete Boolean function.
Similarly, the prime implicates obtained by considering the don’t carecells as 0cells are the prime implicates of an incomplete Boolean function.

9.11.1 Minimal sums of Incomplete Boolean Functions

The prime implicants of the incomplete Boolean function are the terms in the minimal sum. Here the don’t carecells are taken as 1cells if they contribute in forming sub cubes of largest possible along with the other 1 cells; other don’t care cells can be simply ignored. Keeping this in mind the essential prime implicants are identified by first considering only the actual I cells as possible essential 1 cells. A don’t carecell taken as a 1 cell does not qualify to
be an essential 1 cell. These concepts are illustrated in the following examples.

Example
Find the prime implicants and the essential prime implicants of the following Boolean functions using Karnaugh maps.
f (a, b, c, d) = Σm (0, 2, 5, 7, 8, 10, 13, 15) + d c (I, 4, 11, 14)
Solution:
f (a, b, c, d) = Σm (0, 2, 5, 7, 8, 10, 13, 15) + dc (1, 4, 11, 14) Plot the Karnaugh map of the function.






Khannur's Book 

Arunkumar Khannur, Software Testing  Techniques and Applications,
Published by Pearson Publications, 2011
(ISBN:9788131758366; Pages:341 + xxii) 
Follow Khannur 

Khannur's Company 

ISQT Process & Consulting Services Pvt. Ltd., Bangalore, INDIA

Khannur's Software Testing Forum 

STEPAUTO Forum

Contact Khannur 
ISQT Process & Consulting Services Pvt. Ltd.
#732, 1st Floor, 12th Main,
3rd Block,
Rajajinagar,
Bangalore  560010, INDIA
Phone: +91 80 23012511
URL: www.isqtinternational.com
Email: khannur@isqtinternational.com
Skype: arun.isqt 
