So far we can see that applying Boolean algebra can be awkward in order to simplify expressions. Apart from being laborious (and requiring the remembering all the laws) the method can lead to solutions which, though they appear minimal, are not.
The Karnaugh map provides a simple and straight-forward method of minimising boolean expressions. With the Karnaugh map Boolean expressions having up to four and even six variables can be simplified.
So what is a Karnaugh map?
A Karnaugh map provides a pictorial method of grouping together expressions with common factors and therefore eliminating unwanted variables. The Karnaugh map can also be described as a special arrangement of a truth table.
The diagram below illustrates the correspondence between the Karnaugh map and the truth table for the general case of a two variable problem.
Note that values of the input variables form the rows and columns. That is the logic values of the variables A and B (with one denoting true form and zero denoting false form) form the head of the rows and columns respectively.
Bear in mind that the above map is a one dimensional type which can be used to simplify an expression in two variables.
There is a two-dimensional map that can be used for up to four variables, and a three-dimensional map for up to six variables.
Variable B becomes redundant due to Boolean Theorem T9a.
Referring to the map above, the two adjacent 1's are grouped together. Through inspection it can be seen that variable B has its true and false form within the group. This eliminates variable B leaving only variable A which only has its true form. The minimised answer therefore is Z = A.
Note that two groups can be formed for the example given above, bearing in mind that the largest rectangular clusters that can be made consist of two 1s. Notice that a 1 can belong to more than one group.
Verify this algebraically in your notebooks.
Minimise the following problems using the Karnaugh maps method.