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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 care-cells just as the cells wi;h Is were called 1-cells and those with Os were called 0-cells. The prime implicants obtained by considering the don’t care-cells as I-cells are the prime implicants of an incomplete Boolean function. Similarly, the prime implicates obtained by considering the don’t care-cells as 0-cells 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 care-cells are taken as 1-cells 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 care-cell taken as a 1 -cell does not qualify to be an essential 1 -cell. These concepts are illustrated in the following examples.

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.

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