Discrete Mathematics University of Kentucky CS 275 Spring ... - MGNet

Karnaugh maps: The area of a K-map rectangle is determined by the number of
variables (n) and how many (k) are used in a Boolean expression: 2 n4k . Common
arrangements are
• 2 variables: 2$2,
• 3 variables: 4$2, and
• 4 variables: 4$4.
Each variable contributes two possibilities to each possibility of every other
variable in the system. K-maps are organized so that all the possibilities of the
system are arranged in a grid form and between two adjacent boxes only one
variable can change value. Each square in a K-map corresponds to a minterm.
Cover the ones on the map by rectangule that contain a number of boxes equal
to a power of 2 (e.g., 4 boxes in a line, 4 boxes in a square, 8 boxes in a
rectangle, etc.). Once the ones are covered, a term of a sum of products is
produced by finding the variables that do not change throughout the entire
covering, and taking a 1 to mean that variable and a 0 as the complement of that
variable. Doing this for every covering produces a matching function.
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Given a Boolean function f with inputs x 1 , …, x n , make a table with all possible
inputs and outputs. Then create a K-map with the variables on the left and top
sides of the rectangle. Look for 1’s. The rectangle is a torus, so look for wrap
arounds, too.
Example: f: B 4 %B with a corresponding K-map of
x 1 , x 2
00 01 11 10
00 0 0 1 1
x 3 , 01 0 0 1 1
x 4 11 0 0 0 1
10 0 1 1 1
The K-map is colored to try to find patterns in the Boolean expression that can
be simplified. It is quite common to eliminate some of the Boolean variables
using this approach. Use high quality software if you use the K-map approach.
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