Karnaugh Map Logic Minimization
For the operation that corresponds to conjunction (Ù) of propositions we will use the raised dot (×). Sometimes we will use juxtaposition of the variables. Thus xy will have the same meaning as x×y which is the same as x Ù y. This operator is referred to as the meet of "x and y".Karnaugh maps provide a visual representation of the relationship between minterms in an expression. In logically adjacent minterms such as xyz + x'yz (where × = Ù, + = Ú), the connecting variables, x, x', can be factored out and absorbed using Boolean algebra. Truth tables are designed to exhibit the 2N possible combinations for N variables. The Karnaugh map does this as well, but the layout is such that each term is logically adjacent to any orthogonally adjacent term. Examples of two, three, and four variable maps are shown below.
Consider the canonical form example: f = x'y'z' + x'yz' + xy'z' + xyz' + xyz
With the data entered into the Karnaugh map, we make the largest possible groups of adjacent 1's (remembering that the map wraps on its edges). Every 1 must be in a group, overlapping groups are ok as are single 1's. The number of adjacent 1's in any group must be an integeral power of 2, i.e. 1, 2, 4, 8, 16. Translate each group into a product term by including each variable or it's complement if the variable does not change value over the group (recall that the connecting variables would be factored out and absorbed).
Problem
Given the following Karnaugh map, find a logically minimalized expression represented by the map as a function of f.
Incorrect solution Correct solution
