When given a DNF to be minimized, put a 1 in each cell of the Karnaugh map for which there is a conjunction in the DNF. Then group the cells into groups which contain a power of two cells; that is, make groups of size 2 or 4. When we get to larger Karnaugh maps you will be able to make groups of size 8 or 16. Members of groups must to adjacent to one another either horizontally or vertically, and groups must be rectangular in shape. L-shaped groups are not allowed.
Let's use a Karnaugh map to minimize the following DNF:
_ _ _ _
x y + x y + x y
Here is the Karnaugh map with the appropriate 1's placed in cells and then grouped:
The only group that we can make that has the upper right cell in it is the pair in the second column. And the only group that contains the lower left cell is the pair in the second row. Note that we can use the lower right cell in both groups.
Once you have grouped cells, you can determine the minimized logical
expression. Each group reduces to one conjunction in the minimized
expression. A group reduces to a conjunction containing just those
literals that remain the same for every cell in the group. For example,
the group in the above Karnaugh map that contains the two cells in the
second column reduces to the expression "not x" because both of its members
have a "not x" in their DNF conjunctions. The group consisting of
both cells in the bottom row reduces to "not y". Thus the minimized
logical expression is
x + y
When grouping 1's, the object is to get every 1 into as large a group as possible. Sometimes it is not possible to put a cell into a group; in that case, the minimized expression will contain one conjunction for that cell all by itself. As before, if it helps make larger groups we can put a cell in more than one group.
Use a Karnaugh map to minimize the following DNF:
_ _ _ _ _ _ _ _ _
x y z + x y z + x y z + x y z + x y z + x y z
Here is the Karnaugh map:
The reduced expression is
_ _ _
x z + y + x z
The worst case situation when attempting to minimize a DNF is to end up with a Karnaugh map that looks like a checkerboard. In such a case, no minimization is possible.