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Don't-Care Conditions

The 1's and 0's in the map represent the minterms that make the function equal to 1 or 0. There are occasions when it does not matter if the function produces 0 or 1 for a given minterm. Since the function may be either 0 or 1, we say that we don't care what the function output is to be for this minterm. Minterms that may produce either 0 or 1 for the function are said to be don't-care conditions and are marked with an x in the map. These don't-care conditions can be used to provide further simplification of the algebraic expression. When choosing adjacent squares for the function in the map, the X's may be assumed to be either 0 or 1, whichever gives the simplest expression. In addition, an x need not be used at all if it does not contribute to the simplification of the function. In each case, the choice depends only on the simplification that can be achieved. As an example, consider the following Boolean function together with the don't-care minterms:

F(A, B, C) =?(0, 2, 6)

d(A, B, C) =?(1, 3, 5)

The minterms listed with F produce a 1 for the function. The don't-care minterms listed with d may produce either a 0 or a 1 for the function. The remaining minterms, 4 and 7, produce a 0 for the function. The map is shown in Fig. 1-14. The minterms of F are marked with 1's, those of d are marked with X's, and the remaining squares are marked with 0's. The 1's and X's are combined in any convenient manner so as to enclose the maximum number of adjacent squares. It is not necessary to include all or any of the X's, but all the 1's must be included. By including the don't-care minterms 1 and 3 with the 1's in the first row we obtain the term A'. The remaining 1 for minterm 6 is combined with minterm 2 to obtain the term BC'. The simplified expression is

F = A' + BC'

Note that don't-care minterm 5 was not included because it does not contribute to the simplification of the expression. Note also that if don't-care minterms 1 and 3 were not included with the 1's, the simplified expression for F would have been

F = A'C' + BC'

This would require two AND gates and an OR gate, as compared to the expression obtained previously, which requires only one AND and one OR gate. The function is determined completely once the X's are assigned to the 1's or 0's in the map. Thus the expression

F = A' + BC' represents the Boolean function

F(A, B, C)=? (0, 1, 2, 3, 6)

It consists of the original minterms 0, 2, and 6 and the don't-care minterms 1 and 3. Minterm 5 is not included in the function. Since minterms 1, 3, and 5 were specified as being don't-care conditions, we have chosen minterms 1 and 3 to produce a 1 and minterm 5 to produce a 0. This was chosen because this assignment produces the simplest Boolean expression.

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