K Map Don't Care Terms Problems

Example 5: Simplify S = S m(0, 1, 2, 4, 5, 7, 9, 10) + d (3, 8, 15), where ‘d’ denotes don’t care terms.

                                             

Solution: The K-map is drawn as shown in Fig. 2.23. Here, we represent don’t-care terms by Greek letter f, since f may be thought of  as a combination of 0 and 1. We now look for the cells in which the fs are entered, and see whether they can be used to eliminate any variable or not.

      It can be observed that the don’t-care term in cell 3 can be grouped with entries in cells 1, 5 and 7. Hence, in this case, we treat the cell-3 entry f as a valid 1, and it forms a quad with the entries in cells 1, 5 and 7. Similarly, the entry f in cell 8 can be grouped with the 1 in cell 9 to form a pair. However, f in cell 15 cannot be combined with any 1-entry, and hence it is discarded. The final solution now becomes:

                              S = a′d  + b′cd′ + ab′c′₊ a′bc′                                    (2.26)

      It is to be noted that f in cell 8 can be grouped with the 1 in cell 10 also. However, the 1 in cell 10 is already paired with the 1 in cell 2. Hence, we discard the pairing with f in this case.